Leeds has so much choice with bars/clubs. Between headingly, uni and town you have bars and pubs all the way down the main street/Otley run some of the best are..
Hyde park pub is always a good social spot for uni students
Dry Dock is great (especially on a tuesday)
Mook is a cocktail bar in town that does 2-4-1 students (a bit like Font in manchester, but more expensive)
The Terrace/Old Bar in Leeds Union (does the job)
Then clubs.. (all these nights are v popular! especially w. freshers)..
Halo is pretty big amongst the students (open monday and most weekends)
Mezz/Tequila is big on a thursday night (most girls go to mook before)
Mission (thursday night)
Mint club on mondays is popular too, look out for bigger than barry on some tuesdays too
The Space is open most nights
The Warehouse (fridays)
Propaganda (saturdays at the o2)
I've heard Bed is pretty big but I've never been before (lots of my girl-friends love it)
HIFI (recommended, esp on motown wednesdays)
The faversham (tuesdays)
Venues: these nights can cost around £6 to £14 to get in but they usually get quite a few good people to come and play some sets like SBTRKT and Jamie XX, sometimes it can be a bit of a sausage fest, but it really depends on what kind of night you go to (eg drum and bass nights won't have as many girls but deep house will) and you won't find many freshers here..
Mint Warehouse
Mint Club (on nights such as technicolour/louche)
Canal Mills (recommended)
Beaverworks (recommended)
Hope that helped! There're LOTS more places you can go but these are the most well known!