The most popular place to go out is downtown Granville St. It's you typical nightclub street, they close down the roads most weekends and it's full of drunk people. Being from Vancouver, I got tired of it fairly quickly, but I still go down there the odd night for kicks. Clubs close at 2 or 3am. Usually stop serving booze around 2. Music is top 40's, girls are mostly looking for a good time and looking good, but there's a lot of dudes. A lot of people from the suburbs head straight to Granville. For partying, its the highest density area for sure, and I've had many good times on Granville.
Gastown is nice, there's lots of great food, bars, and some clubs around there. It's usually more expensive, but I prefer it over Granville St. Places like the Lamplighter and Fortune can be a pretty good time.
Personally, I usually party more towards East Van. Places like the Waldorf, the Biltmore, and the pubs around tend to be where my friends are at. It's more Vancouver "art scene" if your into that, and they always have live music and dj's. The waldorf has 3 venues in it, so it can be a pretty good time. East Van has a lot of hipsters, but they're easy going. There's a better proportion of Vancouverites at these places. Suburban folks can have issues with pent-up sexual aggression or something. There's some pretty fun after hours places around here, but most places are done at 2-3. Same as everywhere, except for afterhours venues (many of which are illegal.) Sundays would be pretty tame in East Van, though.
If you're into gay bars, head over to Davie St. Places like Celebrities are usually mixed with straight and gay folks.
Really, it just depends what you're into. All these neighbourhoods are within a 10 dollar cab of eachother, and its super easy to catch a cab near bars and clubs. Cover is usually 8-15 dollars, depends what is happening where.
Edit: If you want to visit the quintessential Vancouver nightclub, go to the Roxy. All kinds go there, its smack in the middle of Granville, and its famous for all the pro hockey players that go there to pick up chicks. Its usually voted the best place in Vancouver to get laid, and there's always something going on.