The mechanism
Every mechanical watch is powered by a coiled mainspring inside a barrel; the spring slowly unwinds, releasing energy through the gear train to the escapement and eventually the hands. To run, the spring must be re-tensioned. Manual-wind watches re-tension via the crown: turn it 30-40 times to fully wind. Automatic watches add a rotor, a half-disc weight pivoting on a central axle, which winds the mainspring from the natural motion of your wrist. Both produce identical timekeeping; the rotor is just a power-source automation.
Brief history
The first automatic mechanism was invented in 1770 by Abraham-Louis Perrelet for pocket watches but became impractical because pocket watches don't move enough. The wristwatch automatic was patented by John Harwood in 1923 and became a Rolex commercial product in 1931 with the "Bubbleback" Oyster Perpetual. By the 1950s automatic was the volume default; manual became the high-finish dress-watch convention because manufacturers could remove the rotor and show off the finished movement through a sapphire caseback.
What manual is good at
Visibility: no rotor in the way means the entire movement is visible through a display caseback. Lange, Patek Calatrava, and most haute-horlogerie dress watches are manual for this reason. Thinness: removing the rotor saves 1-2mm of case height. Ritual: the daily winding becomes part of the wearing experience. Classic appeal: pre-1960s vintage watches are almost entirely manual, so manual reads as more "authentic" to the heritage tradition.
What automatic is good at
Convenience: pick up the watch off the dresser and it is already running. Better timekeeping in daily wear: the constantly-fed mainspring stays in its optimal tension band; manuals lose accuracy as the mainspring approaches full unwind. Cheaper to manufacture at scale: which is why most volume Swiss brands (Rolex, Omega Seamaster, Tag Heuer, Tudor Black Bay) default to automatic. The trade-off is movement visibility: most automatics hide the bottom half of the movement under the rotor.
Which should you pick
For a daily-wear single watch, automatic. You will pick it up, put it on, go about your day. For a dress watch you wear occasionally, either: a manual Calatrava or Lange Saxonia is more elegant; an automatic Reverso Tribute is more practical. For a display-back haute-horlogerie piece that you want to admire, manual: the entire movement is visible. Wiki: manual vs automatic goes into the engineering trade-offs in more detail.