Note: This is a pretty old post that I wrote up in response to people claiming that 4th Edition was removing all challenge and risk from the game.

Save-or-die effects are challenging.
SoD effects are not challenging because there is nothing that can be done to stop them, and success or failure hinges on the results of one completely randomized roll. You can argue that there are abilities that can negate SoD effects, but you must either spend a lot of money to get them (thus reducing your overall effectiveness in other areas) or be aware that what you are fighting can do those things. Even then, an item that might negate the effects of a symbol of death or finger of death spell will not work against other SoD effects such as petrification or poisons that deal high amounts of Constitution damage. They are not challenging because there is no skill involved: its pure chance, like gambling. You can sway the odds in your favor all you like, but since a natural 1 always fails there is always a 5% chance that you will just immediately drop dead, regardless of other conditions.
Compare this to challenging encounters, once an encounter starts there are many things that can be done to help determine the overall success of the encounter: one roll will not determine whether the party succeeds or is defeated. Combat encounters are challenging because they largely hinge on player skill, with some chance thrown in.

Traps are challenging.
For most of the same reasons above, traps are also not challenging. Traps are hidden throughout a dungeon, not always in spots where you would expect (pit traps in the floor, for example). Sometimes "clever" DMs will think of an expected space, and put it somewhere else to trick the players (placing a pit trap 10-feet away from the door, instead of in front of it). Aside from the detect traps spell there is no passive way to detect traps, so players have to guess where one might be and make a Search check. If they find it, they must then make a completely randomized roll to disarm it: this again becomes gambling instead of player skill, as the trap will automatically be sprung if the player rolls bad enough. There isnt a way to determine the difficulty and make decisions to affect your results or to mitigate the impact of the trap.
To make matters worse, if you fail you are automatically denied a XP award as well. They are only worth XP if you successfully disarm them and if you fail, the trap punches you in the nuts and thats that. Better luck next time.

Preparing spells is challenging.
Having to select which spells you are going to use ahead of time isnt challenging because you all too often have NO idea what you are going up against in the day. A fighter will generally be very, VERY useful in every combat scenario (and a warblade even more so). If a wizard chooses badly, then he can become utterly useless throughout the course of the adventure. If the wizard chooses well, then he becomes effective.
As someone who has played a lot of wizards (and its my favorite class), I can say that I tended to leave a lot of slots open and choose mainly defensive magic or spells that didnt deal "common" energy types (primarily cold and fire). I tried to pick mostly force spells and things that would allow me to control combat without actually dealing energy damage. This makes it incredibly difficult to make a thematic spellcaster because if you go into an adventure with stuff thats the same energy type, you are useless. Leaving slots open only works so well, as it takes a lot of time to fill them up (and by then it might be too late).
Preparing spells shouldnt be a challenge anyway, as its a fundamental that the class is built on. It would be like saying that fighters should have a hard time picking their weapon because some monsters are immune to swords, or axes, or whatever.

Being useless is challenging.
The normal way that this is worded is actually, "part of the challenge of the game is that you arent useful all the time." Useless isnt challenging, it just means you are useless. Everyone else that has a purpose gets to have fun, while you do not. The usefulness of a character should not come from the class you chose or the class features you picked. Too often spellcasters end up being useless because they run out of spells, or because their array of spells is useless (particularly true of sorcerers and bards). If a character isnt being useful, then the player probably isnt having fun. So I guess being useless isnt useless so much as boring.
There is no challenge with having a useless character. It could be said that its challenging to avoid making a useless character, perhaps, but not ending up useless due to randomized chance.

Elements that only one character can overcome are challenging.
The other side of the coin is sheer usefulness. Situations that only one character can overcome are not challenging because its also essentially randomized. If an adventure has locks or traps, only a rogue is geared to resolve those challenges. If the rogue is dead, lacks sufficient skill, or the party just doesnt have one at all, then the game grinds to a halt. This is not challenging, its a combination of annoying and boring (again).
Similarly, if a "challenge" requires the use of a single, specific spell (such as dispel magic) that the party lacks, the same thing happens all over again. Most DMs will simply remove elements that their players are incapable of overcoming due to lack of resources or powers, but this can still happen in games where DMs are aware of their player's capabilities and planned accordingly, only to discover that they couldnt overcome it anyway (maybe due to character death, bad rolling, or because they didnt prepare that one spell).
Such elements arent challenging because its like a form of gambling: one randomized ability or skill is necessary. This is why in many DM resources it recommends allowing for multiple methods to overcome a challenge.

Dying is challenging.
Again, the way this should be worded is that, "part of the challenge of the game is that you can die." This is typically said in response to people suffering the misconception that you cannot die in 4th Edition. Death is a form of failure (it is not a challenge), but failure is not what makes a game challenging. The fear of failure is what keeps people from just sheathing their swords and ignoring the monsters or whatever. There are numerous ways to fail in a game, with death being the most extreme. This doesnt have to be the only form of failure (retreat, loss of resources, loss of potential reward) are all forms of failure that still allow the game to continue on while still giving the players incentive to try.
Character death cannot be challenging because it is not a challenge that can be overcome. It is just one of many forms of defeat.

So what is challenging?
Currently in 3rd Edition I would say that the most challenging parts of the game are solely combat encounters, or elements within combat encounters. During combat characters have various choices that they can make which generally fall into two categories: attack forms and maneuvering. Character placement can affect the course of combat, and how you attack (whether you are picking a spell or determing how much BAB you are pouring into Power Attack) are all tactical decisions. They typically use randomized rolling, but are greatly influenced by other choices you made (such as how high of a Strength you wanted, whether you are buffed up, and the weapon you are wielding).
This is a problem because D&D typically involves a lot of other dangerous elements that fall outside of pure combat: social encounters, for example, are not challenging because they hinge on a single roll that almost always can be incredibly easily made (especially by a bard).
Trivial things should not contribute to the overall success or failure of the game (such as race, feat, or skill choice). Trivial actions should also not (such as touching a wall or trying to open a door). A good challenge is one that the players can ascertain and make decisions to overcome as opposed to randomly punishing them for otherwise innocuous actions. Players who walk down a hall and get hit for 25 points of damage from a stone block might survive, but they arent going to be happy about it. Players that make a tiefling warlock and are driven out of every village because, "tieflings are evil" will similarly not have fun. It might be "realistic" but its not challenging.
This does not mean that you cannot use ambushes. This does not mean that you cannot use traps. In a good ambush encounter the players might take the first series of hits, but they can still react and retaliate: they arent getting hit and instantly losing, or taking damage only to find that the monsters arent there anymore (and that they cannot be followed or discovered). A good trap allows for everyone to participate in its destruction: fighters can try to block dart attacks while the rogue tries to jam the gears (and other characters try to hack apart other sections of the trap). Sure, you might take damage, but you can at least have a chance to react instead of the DM saying, "okay you take 8 points of damage and make a Fort save from poison," and that no, you wont get XP and that no, there's nothing you can do about it.
Since D&D is a game and not a reality simulator, the fun-factor is important. Too often people forget that really, you're playing a game and you shouldnt feel punished or angry for doing so.

Antioch

"In roleplaying, participants adopt and act out the role of characters, or parts, that may have personalities, motivations, and backgrounds different from their own. Roleplaying, also known as RP to some, is like being in an improvisational drama or free-form theater, in which the participants are the actors who are playing parts, and the audience." -Wikipedia

I think when it comes down to role-playing games that you generally get two camps of thought: people who think that social-RP is the only form of role-playing, and people who think that anything goes when it comes to role-playing. To be fair, I've never heard of an action-gamer refer to die rolling as the "only form" of RPing. It seems to be solely relegated to those who prefer the social variety.

Social-roleplaying is one aspect of RP that occurs when you are talking to another player or non-player character "in-character". The distinction of "social" is important because I try not to give anyone the misconception that roleplaying only occurs when you are talking or character acting. Now, everyone has their own preference on the degree of various roleplaying styles. Some people are content to more or less just play the game without going into a huge amount of character detail. For people that dont game often, or game at conventions, this appears to be a common theme.

What it boils down to is that any action taken by you counts as roleplaying, even if its a simple attack action that is resolved with a die roll. You are playing the part of another character, and thats the point. Role-playing games are still games, no matter how superior you may think they are than board games, card games, or digital games. They need rules. You cant just declare that, "I win", and have that be that. Hence, this is why I think that every situation that actually matters needs rules to help resolve challenges whether social, skill-based, or combat. If you dont include rules for social encounters, then characters can technically defeat them by the player being "really good" at talking, and thats not fair.

I think that social skills actually enhance roleplaying. First of all, they encourage a player who normally isnt good at something to try to do something new. If the character is geared for those kinds of actions, that helps a character further engage in immersive-roleplayi ng (see below). That is, a player who isnt very sociable decides to roll a bard. On top of that, she is also playing a half-elf (+1 racial bonus to Diplomacy checks). A token skill for a bard is Diplomacy, which both makes sense as far as immersive-roleplayi ng is concerned, but synergizes with a bard's role. Now, a first level bard might have around +9 to Diplomacy checks, which is likely far above and beyond the skills of an everyday person (gamer?). Therefore, this bard is a superior animal when it comes to social situations.

If this skill didnt exist, it is very unlikely that Jane Gamer is going to attempt to fast-talk her way out of a sticky situation, because in real-life she's really bad at it. Though her bard might have that sort of thing worked into her background, she cannot accurately portray the role because she isnt good at talking, singing, or casting spells. However, that is the role she wished to take on. The rules help support her in her chosen role, and that assists in immersive gaming as she is able to play her character appropriately.
Immersive-roleplayi ng (immersive-RP) is when you really, really get into character. You talk in character (voice and all), you act in character, you make decisions based solely on what the character would do and not necessarily what you think you might want to do. This can cause problems when people start using character motives as a justification to be a jerk, or to just not play along. Not that everyone who likes to engage in immersive-RP does this, but it can and does happen.

My degree of immersive RPing varies by the group I'm with and the game I'm playing. Generally it is easier to assume the role of things when people cannot actually see you. If you've played MMO games and try to immerse yourself in the game, you probably realize that its much easier to imagine a "voice" for your character despite having to type. It is also easier to assume another race and/or gender, since the people on the other end of the screen cannot actually see you (we've all heard that every female character in World of WarCraft is a 40+ year old, overweight, balding virgin). Visual aids are great for this, and thats why I like using battle mats and miniatures: it is an aid for immersive gamers.

I think that the most immersive character I ever played was Talibah, and only because both DMs insisted it would be crucial to the campaign. It really wasnt, but she definitely had a personality: likes, dislikes, fears, dreams, and all manner of in-depth information. It was fun to kick ideas around and write them down on the extensive background sheet, but almost all of it was never used. Thats a flaw of building up too much information: most of it never sees the light of day, or at best you reference it once (like Talibah's love of dwarven spirits) and then no one cares afterwards.
A final form of RP that I identified while writing this is implied-roleplaying (implied-RP). Implied-RP is the unstated character motivations behind an action. When my fighter attacks a monster, I'm not going to bother explaining the exact reason why he is doing that. Thats what I want him to do, and thats about all the motivation he or she requires. We dont get to know why the monsters are bad guys and why we have to kill them except that it adds to the overall fun of the game. I'm sure there is a reason why, but no one really cares to know, especially in a game where you end up leaving behind the rotting corpses of hundreds of "bad guys". Implied-RP to me helps ignore more questions that is necessary about your character.
You ordered ale, would your character drink ale? Maybe? I guess so, since I did order it. I didnt decide ahead of time whether or not my character drinks, but its safe to assume that yeah, she does.

You chose magic missile as one of your starting spells. Would your wizard really have done that? Its a great spell to be sure, and rather than go into heavy detail about the whats and whys, we can just move past that and play the game.

I use the term roleplaying as a general blanket statement, but I understand that many people do not. Therefore, to help maintain clarity I choose to use various refined terms to help differentiate the many different aspects of the core roleplaying definition. Remember, roleplaying is a catchall term for when you assume the role of someone that is not you, no matter how different. Social-RP is for character interaction, immersive-RP for in-depth details about your character (or when you go the "extra mile"), and implied-RP helps avoid 20 questions every time you do something.

Antioch