Think of the crystal like a tuning fork (which is exactly what some very low frequency crystals are).
If you excite the tuning fork (say by hitting it), it will resonate at a given frequency, but this will die away as energy is lost.
If, however, you add a microphone to pick up the sound, amplify this signal, and feedback to a drive coil attached to the fork, then as long as the phase shift is right, the sound will become self-sustaining.
The frequency is tied to the resonant frequency of the tuning fork, but the power is coming from the external circuit.
Crystals are the same.
The resonance of the quartz block determines the frequency, but to maintain/establish oscillation you need an external drive circuit which picks up the output, amplifies, and then feeds back with the correct phase shift.
The equivalent circuit for a crystal has a series capacitor, inductor and resistance, and a parallel capacitor.
Depending on the oscillator, either the series or parallel mode can be excited.
Have a look at electronics-tutorials.ws (where the diagrams comes from) for an excellent, and detailed analysis. This page also includes a very simple CMOS inverter oscillator circuit (which is referenced in one of the other answers).
(With correct filtering in the amplifier, you can also excite overtones/harmonics in the crystal, but that is probably not what you are trying to do.)
A fully packaged "oscillator" (as opposed to a "bare" crystal) has this circuitry in the package, and will generate an output when power is applied.