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Basically, an oscillator is an amplifier in which some of the output is applied back to the input. This feedback must be positive, and if the gain is sufficient, the circuit will self-drive at a fixed frequency.
There are many oscillators, but two relevant ones are the Colpitts and the Clapp-Gouriet. The Colpitts oscillator uses a parallel resonant circuit, and the Clapp types a series resonant circuit.
Certain crystals, e.g. quartz, exhibit the piezo-electric effect. A small thin section cut from the crystal will possess a fixed natural frequency. Its equivalent circuit indicates that it can operate in either a series or parallel resonant mode.
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