I had a look in the manual and it doesn't give a very good explanation, though the Civilopedia might give a better explanation. Off the top of my head:
There are 2 ways of getting a cultural victory, either civ-wide culture or single city culture.
To get a single city culture victory one of your cities must reach the 20,000 culture point level (this will probably be your capital or a wonder producing city). After reaching 10,000 the culture bar within your city menu will show something like 12,368/20,000. When it reaches 20K you win the game. Presumably if the AI reaches 20K in one of its cities then it wins the game too, but I doubt that would happen unless you let it build heaps of wonders.
To get a civ-wide culture victory the sum of all your city culture points must satisfy the following 2 conditions:
I. It is atleast 100,000 - or 80,000, I can't remember which, but I think it's 100,000.
II. It is atleast double the total culture of the next highest Civ.
You can check your civ-wide culture on your culture advisor with the F5 key.
I've never won with a cultural victory, but from what I hear it is a bit of an anti-climax.
