Thank you, Ferghis - too kind!
Now to confuse things.
There are basically 4 possibilities in any encounter, both sides are ready, the monsters are surprised, the party is surprised, both sides are unready.
Personally, as a DM, I get just a little more complicated than that

, but again, the design isn't precisely specified for these cases.
The example I like to use is, a two-man party is exploring along a North-South hallway, while two wandering monsters are wandering along the East-West hallway that intersects a bit up the way.
Able has a Stealth of +5 and a Passive Perception of 15.
Bob has a Stealth of +0 and a Passive Perception of 10.
The orc has a Stealth of +2 and a Passive Perception of 10.
His pet wolf has a Stealth of +4 and a Passive Perception of 17.
As they draw close to the corner, each makes a Stealth roll as follows:
Able 10, +5 = 15
Bob 12, +0 = 12
Orc 9, +2 = 11
Wolf 15, +4 = 19
The orc's 10 Perception does not hear either Bob or Able. The orc is Surprised, and does not participate in the surprise round.
The wolf's keen ears and 17 Perception hear both Bob and Able. The wolf is not surprised, and participates in the surprise round.
Able's 15 Perception hears the orc's boots on the stone floor, but not the wolf's quiet paw pads. Able is not surprised, and participates in the surprise round.
Bob's 10 Perception does not hear either the orc or the wolf. Bob is Surprised, and does not participate in the surprise round.
So, we have Able and the wolf both taking part in a Surprise round while their partners do not.
. . . . .
What Able and the wolf want to do with Surprise is unclear: but I'd roll initiative at that point and ask what they want to do with a Surprise round.
Able might choose to Charge blindly around the corner ... or he might use a Free Action to motion to the rest of his team to stop and be silent, and declare that he's readying a crossbow bolt on something appearing around the corner.
The wolf might charge around the corner (which would trigger said readied crossbow) ... or he might stop, bristle and growl, alerting his orc that something is up and readying a bite action on something being adjacent to him.
If both Able and the wolf took fairly passive use of their Surprise round, then we're into a normal combat, with normal initiative, and neither side having Surprise: neither side knows quite who they're facing yet, but they're all ready for action.
. . . . .
Now, hypothetically, let's drop the wolf. Able is not Surprised, but the orc and Bob both are. Again, Able could charge around the corner ... or Able might motion to everybody to stop and be silent, and tell me that he's readying a crossbow bolt on something appearing around the corner.
End of Surprise round. In my mental clock, the orc walks about 10' closer to the intersection. The PC's remain aware of the orc; he remains unaware of them; I'd adjudicate that that calls for a
second Surprise round, which you won't find anywhere in any rulebook - but I think its warranted as nobody has actually taken a "that clearly starts combat" action yet.
Bob stops still - "I have my sword out, as usual, but I'm trying to be silent and listen for whatever Able heard." (Minor action for active Perception check). Able might ask what he hears, and I'd tell him "the footsteps are closer to the intersection now," so he might say "I keep my crossbow readied on whatever it is coming around the corner."
In that case, we have the traditional "the (entire) party is aware, and the (entire) opposition is not situation" as described.
At that point, I might well rule that the orc appears at the end of the corridor, and that begins a Surprise round, with each member of the party acting, and the hapless orc flat-footed at the intersection of the two corridors. Being fairly generous about it, I'd let Able choose whether to loose his crossbow bolt, or to try a different Standard action ... but I think a DM would be well within his rights to declare that Able's action in the Surprise round is loosing his crossbow bolt, which he does before the rest of the party do their things.
. . .
The point is, Surprise is almost always relevant, and its almost always something the DM needs to
be thinking about.
When I'm writing encounters, I'll usually detail the approach I expect the party to take, including what the DC's are for each side to achieve Surprise.
I actually do the same thing when enemy reinforcements arrive - my current party, the ranger has a Passive Perception of 18, the Warlord a 15, and the rest of the party a 13, so my "default reinforcements" block reads something like this:
On their initiative in round 2, the reinforcements arrive at the edge of the encounter area. The reinforcements roll Stealth, as follows:
- Stealth DC 19: the reinforcements are placed on the battlemat at the start of their next turn, and get a full set of actions.
- Stealth DC 16: the reinforcements are placed on the battlemat at the start of the Ranger's next turn.
- Stealth DC 14: the reinforcements are placed on the battlemat at the start of either the Ranger's turn or the Warlord's turn, whichever comes next.
- Stealth DC <= 13: the reinforcements are placed on the battlemat immediately, at the end of their current turn, and get no actions.
This results, occasionally, in reinforcements surprising the party, and occasionally in the party getting to act first on the reinforcements rushing into the room, and every now and then an "oh shoot!" moment as just one person gets to react to them before the reinforcements get to go.