You could install a 15 AMP outlet on a 20 AMP electrical circuit but if you try to pull more than 15 AMPS through the 15 AMP electrical outlet you risk melting the electrical outlet and potentially starting a fire.
Most people never even get close to pulling 15 AMPS of electricity out of a 20 AMP circuit so you'll probably be safe but it's best to just put a 20 AMP electrical outlet on the 20 AMP circuit to prevent any overloads or a potential fire.