I bought one Bitcoin today on an exchange at 14.20US$ and sent it to my miner app, the default application I downloaded from Bitcoin.org is that app called the miner? Lots to learn I guess.
Using bitcoins and mining them are different things. It is not that complicated really, if you think of bitcoins as the actual gold coins from the logo.
In oder to keep coins, you first need a wallet. You can even have more than one (Bitcoin PC app, Android client).
Now you need some coins to put into your brand new wallet. The easiest way to get some is to go to a marketplace (MtGox &co) and trade them for other stuff (US$ in this case).
Once you got some coins in your wallet, you can spend them on stuff you like, or donate them to helpful people *nudge nudge*
If you don't have stuff to trade for coins, you can also get some by doing hard mining work. But unlike in the "16 ton" song, it is actually fun!
In order to start mining, you first need to get a mining tool that suits you (one of the many miner apps).
The you can saddle your mule (no analogy, sorry) and go rinding out as a lone prospector, trying to dig up some gold (a valid block, worth 50 BTC).
However, since the gold has become scarce over time and there are lots of miners out there, your chances of finding the mother lode on your own are slim.
Instead, you can join a bunch of other friendly miners (a pool) and help them at their mine (deepbit.net &co).
Once you or your mining buddies actually manage to find some gold (the valid block), everybody gets a fair share.
This way you get less gold at once, but you get it at a steady rate. Much better for the impatient.