It depends a lot on the age of the bike.

If you have an old bike, you don't need much more than a dumbbell wrench:
In addition to which you probably want a chain tool, a spoke wrench, a screw driver, tire levers, and a pump.

A cone wrench or two can be very useful, and a 15mm open end wrench is needed for pedals.

If you're working on newer bikes, you need allen wrenches and a crank puller.