LiPo stands for "lithium polymer" but for the sake of hobbyist robotics you can consider them Li-Ion batteries (no memory effect, high power to weight ratio) with the added feature of high charge/discharge rates.
As far as the "C" goes, it's shorthand for charge/discharge rate. Lets say you had a battery that had a capacity of 1Ah. If you charged (or discharged) it at 1 amp for 1 hour that would be considered 1C. 2C means you get 2 amps for 30 minutes, 3C means 3 amps for 20 minutes. So for a LiPo battery with a charge rate of 20C, you could get up to 20 amps for 3 minutes. (This is all in an ideal system, in real life you have to take into account voltage curves, inefficiencies, etc.)
As far as selecting a battery is concerned, it's the same as any other part: Calculate what you need then get 150% of that.
[opinion] I think that switching to LiPo batteries for the sake of saying "I've got LiPo batteries" isn't a good idea. If it isn't broken, don't fix it.[/opinion]
