The tools you need do depend on the bike.

The most essential tool is a pump. You need to be sure the pump fits the valves on your inner tubes. The common valves are presta and schraeder. Schraeder valves are like the ones on car tires. A floor pump with a built-in pressure gauge is really nice. Bike tires tend to lose pressure faster than car tires and you really want to keep the bike tires pumped up properly.

After that you will want to be able to fix flats. You'll need "tire irons" to get the tire off the rim. You need to remove the wheel from the bike, too. You might have bolts on the axles, for which a big adjustable wrench is OK or better to have a wrench of the right size. Or you might have quick release levers which don't need any tools.

So a set of (probably) metric open/closed end wrenches is nice, and then also a set of (probably) metric Allen wrenches (hex keys) is pretty much essential. Look at the various fasteners on you bike and get what you need to loosen/tighten them!

Get a basic book on bike maintenance too!