Mechanical watches rely on the period of the balance wheel to allow the escapement to advance the mechanism at a regular pace. When a watch is designed, and then built, a hairspring is used that will have a period appropriate to the watch.
The most common rate of a mechanical watch is 18,000 beats per hour, although there are other rates. The watch's rate is the ideal period of the balance wheel, which is essentially a pendulum whose period is dicated by the length of the hairspring. In most vintage watches, the length of the spring can be controlled by a regulator. The regulator turns around the center of the balance and moves two regulator pins along the begining stretch of the hair spring. The spring passes between the two pins. Moving these pins makes the hairspring effectively longer or shorter and thus changes its period.
In most watches, the full range of the regulator will not result in a change of more than two or three minutes a day, give or take.
Mechanical watches do not have perfect rates though. If everything else is functioning well, the intent of the regulator is to evenly distribute rate errors, created by things like the watches physical orientation throughout the day, or temperature changes, so that on average the watch reads well over time. But it is a relatively fine adjustment.
What if a watch runs very, very fast? Like gaining a minute or more in an hour?
A hairspring problem is the most common cause of a watch running much too fast. There are others. For example a mainspring that is too strong can drive the escapement too fast. The escapement may be damaged causing it to sometimes slip. Or a wheel may be missing a tooth causing it to jump ahead as the bad spot comes around.
What if a watch is very, very slow, such as losing minutes an hour?
Interestingly, under normal conditions where all the parts are correct, a watch can not physically run very, very slow. Again, the period of the balance dictates the frequency at which the escapement allows the mechanism to unwind. If a watch is behind by an hour at the end of a day, it is almost certainly because it is actually stopping completely at some point and restarting without being noticed. A watch might do this once for a long period, or several times during the day. A huge array of factors play into problems causing a watch to stop, some of them causing just slight pauses each hour, which may be hard to notice.
The long and short of it is that vintage mechanical watches have imperfect rates, rates which also vary slightly as they run. Small rate errors can be accounted for, through the fine adjustment of the regulator. But larger errors indicate a more serious trouble. A very fast watch probably has a hairspring problem. A very slow watch is probably stopping.