I saw an explanation posted above, got about 2 lines into it, then said 'you know what, I'm not that interested' and bailed
OK, here's the clearest way I can explain it.
- Up until some time in the '60s or '70s, all watches were mechanical, just like a little Grandfather's clock. This illustration explains it
A mechanical watch has a mainspring which is wound - and then unwinds - to keep the gears moving and keep time. The hands of the watch are connected to the gears. The pinion part looks like a little pendulum when the watch is running.
Mechanical watches can get super-complicated when you have it change the day and date and do chronograph and lunar functions.
A mechanical watch must be wound by hand. But if you add a rotor which swings with the wearer's movements, it'll wind the watch automatically. Hence the automatic. In the illustration below, the rotor is the half-disc at top.
Here's a watch movement with a rotor
If you wear it everyday, it will keep itself running. They make watch winders that move the watch when you're not wearing them, if you care about having the watch keep running (I don't).
A quartz watch uses a quartz oscillator chip in an IC package, and a battery, to move the gears that are attached to the hands and date.
A quartz watch doesn't have most of the parts that are needed for a mechanical watch. In some quartz watches, a solar panel charges the battery. That doesn't make it an "automatic" watch.
Quartz vs. mechanical watches are analogous to internal-combustion-engine cars vs. electric cars. They both look the same externally, but one has an engine or mechanics, and the other just a battery and chips.