Crystals below their resonant frequency appear mostly capacitive. Above their resonant frequency, they appear mostly inductive. At their resonant frequency, they appear mostly resistive.
Re-draw the Pierce oscillator three times, replacing the crystal with one of those components. It may help you understand how it works.
Parallel resonant crystals are actually specified a little bit under the fundamental frequency. This makes the crystal appear a bit capacitive at the spec'd frequency. The additional capacitance adds a bit of additional phase shift to help the oscillator start and run.
The amplifier's input sees a bigger signal near the crystal's fundamental (resistive, typically under 100 Ohms ESR). The smaller off-frequency signals are diminished or blocked, so a signal at the fundamental frequency grows stronger (after being amplified) and dominates.
Push someone on a swing. No matter how hard you try, the swing really will only move back-and-forth at some fundamental frequency.
Imagine a crystal as the surface of the water. Now send ripples (waves) across that surface. The ripples move the surface up and down, effectively bending the surface. The crystal too bends as it vibrates.
Bending can be caused by applying an electric field to a quartz crystal, but also the bending itself creates an opposing electric field in the crystal lattice. At rest, these forces are balanced, and the crystal has no charge.
Which is easier to vibrate with your hand: a 12x1 inch ruler, or a 6x4 foot sheet of plywood? Obviously the smaller ruler can be vibrated faster!
Crystals are the same. Their dimensions determine their resonant frequency; smaller and / or thinner crystals vibrate faster. This is also what limits the fundamental frequency of a crystal: crystals get too small or too thin to accurately process by mechanical machining or chemical etching at higher frequencies.
At really low frequencies, crystals become so large or thick that it takes too much power to make them bend; hence a tuning fork crystal design is used for low-frequency 32.768 kHz timing crystals.
Crystals can actually oscillate at more than one frequency. These are the overtones at multiples of the fundamental, but they tend to be weaker than the fundamental. It is possible to design a circuit to cause a crystal to oscillate at an overtone, typically the third or fifth. Typically crystals over 40 MHz are designed for 3rd or 5th overtone, not the fundamental, so carefully read the specs before purchasing!