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Short answer: A crystal is a two-terminal network which has a complex impedance between it's two terminals. At a frequency near "series-resonance", the impedance across the two-terminals becomes real only, and fairly low. It is possible to build an oscillator (without extra capacitors) based on series-resonance of a crystal.
Most crystals are operated on what is called "parallel resonance", where an external capacitance is place in parallel with the crystal terminals, where the external capacitor resonates with the effective inductance of the crystal. Usually, to create an oscillator, you need to create either a voltage step-up, or a phase reversal, so the external capacitor is split into two series capacitors, creating a three terminal network which is incorporated into the oscillator circuit.