I used to use a Logitech c920 and Logitech's included webcam software. It's a great solution if you can make it work, but using an actual webcam has two distinct problems:
1) It uses resources. Unless you have a totally diesel computer, you run the risk that by recording with the same machine you're also asking to play a game while recording THAT, it might slow everything down.
2) If you're playing a fullscreen game, you can't be 100% sure it's recording properly. There won't be any onscreen indication that the webcam is rolling.
On the other hand, it's cheap, mounts on the monitor, and doesn't require anything fancy. Personally, I use a Sony CX240. Excellent quality, even if it's lacking a few niceties (mostly with the interface).