A bitcoin is an alternative currency based on crypto. Each new coin "mined" by someone takes more work than the last - it was an attempt to solve the problem of someone having a printing press devaluing them for everyone.
The short answer to how long is "much too long" with a normal computer - that's been true now for a couple years. Then people doing it went to specialized GPU cards that were maybe 100x faster for awhile - the power draw became significant (not to mention the upfront cost of GPU cards with a thousand or more compute cores). Now, people like butterfly labs and others sell a special machine that will do many giga-hashes (the underlying operation is SHA) per second, those are just hitting the market now.
Governments don't like competition, especially in currency - lots of bitcoin exchanges have been shut down (and people have lost their digital money), and there is actually an issue of theft via malware.
The prices go all over the place vs FRNs. Some speculate. And the Winklevoss brothers (of trying to steal Facebook fame) have started an ETF to trade them. That should tell you quite a lot - some of the most famous, spoiled-brat-dishonest people on the planet plan to control them.
Here's an ars article on someone with a butterfly labs box mining them. He did OK, but that was then - remember it's getting harder to do with time, and ever-faster dedicated machines, and farms of them are coming online every day - you're not going to do all that well competing with that as they find all the "easy ones" - pretty much already.
http://arstechnica.com/gadgets/2013/06/how-a-total-n00b-mined-700-in-bitcoins/
This box is perhaps a few tens of thousands of times faster than your computer, FYI, at doing the necessary math. As you'll find out if you do some digging, there are farms of 1000's of even faster boxes you'd be competing with. As they say on Slashdot - GoodLuckWithThat - I'd be doing it myself were it worth it, but my fastest machine (an i7 with 2k CUDA GPU cores on two Tesla cards) isn't even in the running anymore - and it draws >500w just to run, so it's just not practical here off the grid to leave it on all the time.
The situation is well into the space of using custom designed ICs for this now. Unless you want to go there...it's not going to be worth it other than as a novelty. The issue of course, is that the guys who make these keep most of them - the people who get rich in a gold rush sell the shovels, usually, but in this case they can sell the shovels, while keeping the big machines themselves and do well off both.
