The Colpitts, Pierce and Clapp oscillators comprise ...
- a resonant circuit consisting of two series capacitors and an inductive element (an inductor, or a crystal, ceramic, MEMs or series LC resonator operating on the inductive side of resonance)
- an amplifier arranged around it to maintain oscillation
The main distinguishing point about the Pierce oscillator is that it has the mid-point of the capacitors grounded. This allows the amplifier to be a two-port inverting type like a logic inverter gate, ideal for integration with logic devices, rather than a three terminal device like a transistor that needs biassing.
A Colpitts oscillator always uses an inductor. When operated with a resonator in the inductor position, it's called a Clapp oscillator.
Both Colpitts and Clapp oscillators can be arranged with different terminals grounded, so the transistor is used in 'common base', 'common collector', or 'common emitter', with the first two appearing to be more popular.
There are incidental differences in the use of the oscillator types. Fine frequency/phase stability of a logic clock is not often required, so the amplitude control of a Pierce clock is done by allowing it to slam against the rails, giving it poor power supply and amplitude noise rejection. A radio receiver OTOH needs quiet signals, and will use a transistor biassed so that it goes into emitter cutoff each cycle, far quieter than the alternatives.