Hmm, I just had a new player ask me this the other day -
Let me copy out of my reply email to him - Reference your PHB for the appropriate Tables, I don't think they'll format well, so I'm leaving them out.
"Think of the difference between spells known and spells per day this way:
For Example - using a Sorcerer (and the following tables) who is 7th level-
(TABLE: Sorcerer Spells Known)
If you look at the above table, you see the sorcerer knows two 3rd level spells. Let's say he knows Fireball and Fly.
(TABLE: Sorcerer Spells Per Day )
Now if we look at this table, we see that he can actually cast 4 3rd level spells per day (more if his charisma score gives him bonus spells of 3rd level). Since a sorcerer does not have to pick his spells for the day in advance (like a wizard or cleric), he can cast any combination of the spells he knows per day.
For example:
Fireball, Fireball, Fly, Fireball
or
Fly, Fly, Fly, Fireball
or
Fly, Fly Fireball, Fireball
or
Fireball, Fireball, Fireball, Fireball
etc.
This makes the sorcerer fairly powerful in combat compared to a wizard because he can pick what he wants to cast right before he casts it. However, there is a down side. The Spells known are never able to be changed.
Let's say the same sorcerer was 6th level, meaning he knows one 3rd level spell, and he picked Fireball. When he advances to 7th level, he gets to pick an additional spell. But he has not found Fireball to be of much use! He can never "unlearn" a spell, he will know Fireball from now 'till whenever, he can never get rid or trade in that spell for another one known.
This is where wizards have more flexibility, they memorize spells each day, but can pick whatever is in their spell books to memorize. There is no limit (other than cost) to how many 3rd level spells they can have in their books, they could have all of them! But they have to pick what they will have available at the beginning of each day, so they might not have picked what they might need. They also get the higher level (and thus more powerful spells sooner).
Here is the Wizards Table -
(TABLE: Wizard Spells Per Day)
There is only one table, no spells known because if there are 20 3rd level spells, he could possibly have them all written in his spellbook.
Compare a 7th level wizard to the 7th level sorcerer in spells they can cast per day. The wizard table shows 4/4/3/2/1 compared to the sorcerer 6/6/6/4. The wizard has less spells, but has access to at least one 4th level spell. But at the beginning of the day, the wizard has to pick his spells. He has to write on his sheet "Fireball, Fly" If he casts his Fireball, it's gone for the day. If he really needs to cast another one, he can't! So it's important to choose which spells you have memorized each day.
Clerics work similar to wizards, except they have no spellbook. At the beginning of each day, they pray for spells. But like wizards, they have to select what they have for that day ahead of time.
Here is a Cleric table -
(TABLE: Cleric Spells Per Day)
*In addition to the stated number of spells per day for 1st- through 9th-level spells, a cleric gets a domain spell for each spell level, starting at 1st. The "+1" on this list represents that. These spells are in addition to any bonus spells for having a high Wisdom"
So, as a Wizard, so will have a spellbook. In it are written the spells you know. Under the Wizard class, it tells you how many you start with. Everytime you gain a level, you get to choose 2 more spells of any level you will be able to cast (including from you new level) and write them at no cost into your spellbook.
So using your 4th level Wizard, he will have his basic spells, 2 more for getting to be 2nd lvl, 2 more from getting to be 3rd lvl and 2 more from getting to 4th. These will be in his book at no additional cost. See the spellbook section in the PHB, I believe it's around page 155 or so.
Now, If you have an amount of funds, you could ask the DM if you can have purchased more spells. You could buy a spell on a scroll, memorize it (destroying the scroll) and write it into you book. Not only will this cost to buy a scroll, but also it will (by the PHB) cost an additional 100gp per page to write it into your spellbook, and take an amount of pages equal to twice the spell level. (3rd lvl spell take 6 pages, only 0 lvl spells take 1 page each).
A book holds 100 pages. You will start with one with your starting spells (which will start to use up pages).
Does this clear things up any?
[edit - In the above examples, it does not address additional spells gained for you bonus in your casting stat, I was trying to keep it simple - see your DM to discuss that]