Quartz Crystals and other Oscillators
Many electronic devices need a fixed clock or frequency source to work properly. A quartz crystal is made for this purpose and comes as 2-legged unit in a metal case.
A quartz crystal works very similar to a tuning fork - generating a "tone" when hit and thus deliver this tone - a frequency - in a very stable manner. This tone is usually not audible, far above the perceptable frequency range of the ear. Quartzes are available from several kHz (might be audible) up to several 100 MHz. The frequency of a quartz is in fact a mechanical oscillation - directly translated into an oscillating voltage. Hitting a quartz crystal can cause disturbances or in very sensitive equipment crash it (very uncommon).
Most important is the frequency of the quartz. If the frequency is too high the circuit runs too fast - the circuit is overclocked. The same happens if the frequency is too low - the circuit runs slower than expected. You can try to replace quartz crystals in toys which create sounds - they'll be pitched up or down.
The second important property is the accuracy. The quartz in a wrist watch must be very precise - or else the watch will run too fast or too slow. This also applies to Arduino boards - if you need an accurate clock, use an additional crystal which is precise. (The built in quartz is not very precise).
Many applications don't need high Accuracy of the Frequency and can use cheaper oscillators, for example a "ceramic resonator" - this isn't made of quartz but ceramics instead. Their frequency is more widely varying.
A quartz has 2 connections which are internally connected to 2 sides of a piece of precicely shaped quartz. This fragile construction is protected by a metal housing which also shields the crystal from electric influences. The 2 legs have no polarity - it doesn't matter in which direction they are connected.
4 legged quartz?
apart from 2 legged crystals which are found in most devices there are rectangular shaped metal-ish shiny boxes with a printed frequency marked on them. They contain quartz crystals and additional circuitry to generate a signal directly suitable as clock source. They need power supply to run - thus 2 legs more.