You can use the webcam Pahuffman listed to record videos as well as stream live if you like if your computer is beefy enough.

In either case, you're also going to need to work lighting and stands into your budget as well as a mic or two depending on how much game sound you want vs your voice. Don't forget stands to mount your camera(s), a bunch of long USB cables, a powered

Basic list of things to do this mostly right:
* Camera(s)
* Boom stands for the camera(s)
* Lighting with diffusion
* Mic and mic stand - more if you want quality audio from the game and your voice.
* Long USB Cables
* Powered USB Hub

More or less depending on your budget and ultimate goals for production quality.

That video is 3 cameras. If you're doing a game like DI or SW and want to show detail of the smaller LCD AND your face, you need 3-5.

Of course, OP just asked "What camera" and might know all this other stuff.