Yes you can put a 15 amp plug on a 20 amp circuit, but why would you want to?
You can also put a 20 amp plug on a 15 amp circuit, but you run the risk of burning down the house and everything in it. Oh yeah, and it's against pretty much every code for that reason!
Some people use 10 guage, but 12 Guage wire will handle both in a home setting, but the breaker might have to be changed to match the plug.
Hope this helps.
JD