An automatic wristwatch is a mechanical wristwatch with a self-winding mechanism. In other words, one does not have to wind the crown periodically to keep the watch running. A manual or manual wind watch must be wound by hand, using the crown, usually every day, to operate continuously.
If one were going to own only a single watch, and wear it every day, an automatic would be a good choice, since the watch will be worn consistently enough to stay wound – the owner would never need to manually wind the watch, and would only need to adjust the time to compensate for drift and at changeover to daylight/summer time and back. (In fact, several early automatic movements dispensed with the crown and moved the time-setting mechanism onto the back, under the theory that the mechanism would only be accessed infrequently. This turned out to be a marketing flop – people liked the look and easy accessibility of the crown.)
For this reason, most commonly seen watches with more than a simple date window use automatic movements – this includes “triple date” calendars, annual calendars, perpetual calendars, and any of these combined with moon phases. With few exceptions (oddly enough, these seem to be more expensive watches), most manual wind watches have simpler calendars, although they may include other complications like chronographs.
One caveat about automatics – if you have more than one watch that is worn regularly, the automatic winding advantage is lessened – the automatic may stop if not worn often enough. With some calendar mechanisms and other complications this can add an inconvenience factor to resetting the watch.
Finally, since frequently worn automatics are usually at or near a full state of wind most of the time, one may get the impression that they can be adjusted to be more accurate and consistent over the course of many days. This, in fact, is not necessarily the case, as a manual-wind watch that is wound consistently once per day can be tweaked so that the day to day variation is very small. In short, there is no definite performance advantage to an automatic – it is mostly a convenience.