To keep it simple, lets forget about the wire gauge for this discussion.
If you have a 20 amp breaker, you can use a 15 or 20 amp receptacle.
Yes!
If you install a 15 amp outlet on a 20 amp breaker, you have under-protected the outlet by 5 amps. Besides being illegal, if you plug a device capable of more than 15 amps of current draw, the ampacity of the 15 amp outlet will be breached and melt down will start to occur in the outlet because the 20 amp breaker will not trip. Then you will have fire. Very bad.
Yes! For anyone that needs a reference, please check table 210.21(B)(3) of the NEC code, a 15A branch circuit can only have a 15A receptacle attached. A 20A branch circuit can have either a 15A or a 20A receptacle attached.
The danger is that a 20A device could be plugged into a 15A circuit, the breaker fail and not trip, and start a fire.
A 15A device plugged into a 20A circuit can draw 15A all day long and all will be well.

A 20A device plugged into a 15A circuit would be bad, in the event the breaker did not trip.

FYI--most commercially made receptacles 15A and 20A receptacles have the same guts, just different face plates.
1. To play it safe, always replace outlet with same amperage rating as circuit breaker.
2. Never change out a 15 amp breaker with a 20 amp breaker (unless you change the wire gauge).
3. Install wires according to instructions that came with the outlet.
4. Better yet, have an electrician do it.
Yes! #4 is fantastic advice, thank you Wayne!
I don't want to keep beating a dead horse here, but, I want to make sure the correct information is relayed.
All of this being said, 20A receptacles are installed all the time on 15A circuits by audiophiles. Just be aware, it does not meet code. That's all I wanted to get across.
Peace,
Lee