Almost every watch in the world keeps time in one of three ways. Knowing the difference is the single most useful thing for understanding what you are wearing or buying.
A quartz watch runs on a tiny battery that sends current through a quartz crystal, making it vibrate exactly 32,768 times a second. That steady beat is counted to move the hands. Quartz is the most accurate everyday technology, cheap to make, and needs almost no attention beyond a battery change every few years. The seconds hand usually ticks once per second.
A mechanical watch has no battery. You wind the crown, which tightens a mainspring; as it slowly unwinds, a train of gears and an escapement release that energy in tiny, even steps, swinging a balance wheel back and forth. The seconds hand appears to 'sweep' smoothly. It must be wound regularly, is less accurate than quartz, but can be serviced and kept running for generations — which is why collectors love it.
An automatic is a mechanical watch with a weighted rotor inside that spins as your wrist moves, winding the mainspring for you. Wear it daily and it never needs hand-winding; leave it on a shelf and it stops after a day or two and needs a wind to restart.
Neither — they serve different wishes. Quartz wins on accuracy, price and convenience. Mechanical and automatic win on craft, longevity and character. Many people own both.