Does the Bitcoin client generate bitcoins automatically? I mean, you just need to install the software, and with the time, you will get lots of bitcoins?
Not really, there are a few issues you have to know about:


In order to generate Bitcoins, you have to be explicitly mining them. The standard client does not mine coins, it stores the keys you use to access them. You need to run a so called miner (like the GUIMiner) and you probably want to join a mining pool (like, Slush's Pool).
Even if you are mining Bitcoins, you probably won't make too many coins unless you have some good hardware. You should check out what hashing speed your hardware has and then punch the hash speed in a dedicated calculator.
If you want to mine coins by yourself (without pool), the chances of getting coins are pretty small, again, check the calculator and look for "Time to generate one block".


So in short, you need specialised software and hardware to mine coins, but it's not hard to get started.