Does the program I downloaded (bitcoin-qt) mine bitcoins?
Yes, but not efficiently. In fact, CPU's are so bad at mining bitcoin that the option was removed from the user interface. However, the code is still there (though mostly as a reference implementation), and you can go to Help > Debug Window > Console and type in "setgenerate true" to turn it on.  Type "setgenerate false" to stop.

You can also give bitcoin the option -gen at the command line.