RS Components is a good source of laser diodes, some of which are theoretically capable of burning foam, paper, thin wood etc. be prepared to do lots of googling though. ;)
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That really depends on what you want to cut. If you want to cut anything thicker than tissue paper, it's not going to be cheap. A cutting laser has to be quite powerful, or else it'll just burn whatever you're trying to cut. It has to cut through extremely fast and move away from that spot, so the laser you need will probably be expensive.
For metal, the laser just heats the metal and a blast of oxygen, argon, carbon dioxide or other gas burns or blows out the molten material. For thin stuff, 100+ mWatts (paper-type thin). Don't forget that most high power lasers need cooling or they burn out.
Best bet is to find a laser cutter that can do what you want and check the specs on it for what they use for power.
For sources, try E-bay. Not too many manufacturers are going to be willing to sell one diode, unless it's a replacement for a machine they produce.