Alt-Sachsenhausen

Alt-Sachsenhausen, a part of the suburb Sachsenhausen south of the Main river, is famous for its bars and Kneipen (a German type of Bar, really) serving the "national drink" Ebbelwoi (local dialect for "apple wine", sometimes spelled Ebbelwei). However, these days it's mostly for tourists. A better option in Sachsenhausen is along Textorstrasse, a two minute walk south, where you can still find a row of authentic places catering to locals.

Not so famous like "Alt-Sachs", but also well known, is Bornheim (located in the north) which has also some biergarden-like applewine establishments at 'Berger Straße' and the surrounding area.

Mid-range

Cafe & Bar Celona Holzgraben 31 (near Hauptwache). Tel +49 69 13886709 Daily 9AM-1AM, Fri & Sat 9AM-2AM. Spanish-style bar in the heart of the city. Popular, attracts a younger crowd, good for meeting people. Very crowded around 5PM-7PM. Also serves a variety of Spanish dishes. ?7-20/person.
During the summer, you can enjoy one of the open air rooftop bars with beach atmosphere

Clubs
There are many clubs in Frankfurt.

Cocoon Club - probably the most famous and extraordinary techno club in the world. Definitely worth a visit, even if you don't like the music, due to the amazing modern interior design. Carl-Benz-Strasse 21-35, Fechenheim.
Jazzkeller - since the 50ies the Jazz meeting point, Kleine Bockenheimer Strasse 18a T: 069 28 85 37.
Velvet Club - new club in downtown Frankfurt with very nice interior design. Great place for house and electronic music lovers.
Kingkamehameha Club - one of the most famous clubs in Frankfurt. Dress to impress!
Monza - small techno club in the heart of Frankfurt.
U60311 - famous underground techno club in downtown Frankfurt. Hotspot for the world's most famous techno and house DJ's.
Living XXL - club in the basement of the European Central Bank. Very famous for its after-work parties on Wednesdays.
Odeon - beautiful club close to the "Konstablerwache". Very popular among students.
Clubkeller - small place on Textorstrasse that has a good mix of indie music and a nice atmosphere.
PULSE Club - popular gay club in Frankfurt with two dancefloors.
Batschkapp - THE place for rock and alternative music. One of the oldest clubs in town.
Nachtleben - nice Bar/café close to "Konstablerwache" with a small concert hall in the cellar (mostly rock, punk, metal).
Sinkkasten - one of the oldest clubs in Frankfurt. Has a mixed programm from Modern Rock to 60's, 70's, 80's Disco Music, concerts...
Kingkamehameha Beach Club: artificial beach, open only during the summer, not at the same location as the regular Kingkamehameha Club!
Along the Hanauer Landstrasse and around central station there are a lot of clubs.