There are two basic cases for a mechanical watch running fast, and determining which case it is depends on how fast it's running. Of course, like anything, the two overlap. If it's ahead by less than 20-30 seconds a day, then this is something which can be corrected by regulating the balance. If it's running faster than that, then you're in to the second case, which is that something is wrong with it.
So, if it's running not more than 30 seconds a day fast, you can adjust the regulator with a few simple tools. You'll need a way to open the case, which may be held on with screws, or may be a one piece setup that screws into the main body of the case, requiring a special wrench. And, in most cases, you'll need a good quality small screwdriver to do the adjustment. Typically you'll use a .80mm screwdriver for this. On a fairly typical ETA movement, like the widely used ETA 2892, there will be a regulator screw on the balance cock. On a watch with a display back, this should be easily visible. It's a small screw, with a sort of "V" shaped bracket around it, and a series of hash marks on the balance cock, with a "+" at one side, and often a "-" at the other. The screw can be turned, using the orientation of the slot lined up with a hash mark, to either plus or minus directions to make it gain or lose time as needed. How this works is that the screw is asymmetric, and rotating it slightly changes the position of the "V" shaped bracket around it, which in turn rotates a collar that moves the curb pins around the hairspring, effectively shortening or lengthening the hairspring, which gives it a slightly shorter or longer return on the oscillation. Older watches will often have a small lever that accomplished the same purpose, and in most cases it simply slides one way or the other. Some will have a screw set in the side of the balance cock which pushes on this lever, along with a guard spring that pushes against it the other way, allowing the lever to be very accurately fixed in place through the dual action of the screw from one side, and the "swan's neck" spring from the other. Old Omega movements often have this sort of adjustment, and it's considered a very high quality feature.
So, if you're running close to accurate, and the regulator system isn't already set a maximum slow, then you can slow it down this way. The exception is what's called a "free sprung" balance, which does not have a system for altering the effective end location of the hairspring and instead relies on adjustments of the balance to retain accuracy. Don't attempt to regulate these yourself unless you're a skilled watch maker. I believe all current Jaeger-LeCoulter models are free sprung, as are Rolex, and the new Omega Co-axials, as well as many other high quality watches. Generally speaking, a free sprung balance is a superior system since, once set up and regulated professionally, it's not going to change. But, it requires more skill and time during manufacture or repair to get it properly dialed in to begin with.
Very inexpensive watches, such as low cost Chinese made automatic movements, will usually lack a mechanism to easily move the curb pins, but will often, inexplicably, have "+" and "-" marks on the balance cock where there isn't a regulating system. Those which do have a provision for moving the curb pins will generally just have a sliding lever such as was common on low-to-mid range Americans and Swiss watches of the 40s-70s. For ones without a regulating system you must move the curb pins manually, using a steady hand and being careful to not touch the hairspring. There are two tabs sticking out from the center of the balance cock. One is the anchor stud for the hairspring, and the other is the hairspring curb pins. The curb pin tab can be identifies by the two small gold colored studs visible on it. What you're looking at is the back of the curb pins, which are set directly into the tab. This tab should slide around the end of the balance cock, with a total range of movmement of about 30 degrees. Moving it towards the other tab, clockwise, will slow the movement, and moving it counterclockwise will speed it up.
Regulating this way is best done will small adjustments, and a day of running in between to get an accurate idea of what the adjustment has accomplished. Bear in mind that other factors, such as how fully wound the watch is, whether it's been left overnight sitting face up, or on its side, will change the run rate, too, so there will always be at least a few seconds a day "slop" in there, even on a good quality Swiss made chronometer grade movement. But, with patience and small steps, you can get a mechanical automatic regulated to run quite well. Wear and general condition of the movement will alter the run rate, too, and as long as it's minor and can be accommodated by the regulation system, the watch can be brought back in to an accurate running state using this method.
However, beyond this 20-30 second a day zone is where you're looking at an actual problem. More often than not, it's simply a matter of the watch needing a thorough cleaning and re-oiling. General dirtiness of the movement will make the watch run faster by robbing power out of the system. This may seem counterintuitive that less power running though the system would make it run faster, but the explanation is simple. The balance of the watch is an oscillating system, a wheel with a spring which returns the wheel back the way it came if it's rotated. The escapement kicks the wheel through the impulse jewel, which applies force to the hairspring, which then reverses the direction of the balance wheel, which then comes and kicks the escapement on its way past. During this time, the escapement has the mechanical system "locked", so the balance is controlling the rate at which everything runs by unlocking the escapement at a predictable rate, and that rate is determined by the oscillation behaviour of the balance wheel-hairspring system. Every time the escapement gives the balance wheel a kick, it locks, and every time the balance wheel spins back past center, it unlocks, gives the balance a kick along its way, and locks again, and this action repeats around 28,800 times an hour for most movements. If there isn't enough power being delivered by the escapement, the balance doesn't get too far before the hairspring returns it, resulting in a low amplitude oscillation that happens more than 28,800 times per hour, and the watch runs fast*. In the case of an extremely dirty and gunked up movement it can run up to several minutes per hour fast.
So, running fast beyond the 20-30 seconds per day window means it's time for a cleaning, and possibly some replacement components. This is best left to a professional, and requires a few hours of highly skilled labor, at the very least. You can certainly learn to work on your own watch, but it will take a lot of practice, a lot of mistakes made, a lot of patience, and some specialized and expensive tools to be learn to clean and rebuild a watch correctly.
High magnetic fields can also cause strange and erratic running, including running amazingly fast. So, if you've handled powerful magnets, or worked around strong magnetic fields, this is also a possibility. This causes a watch to run fast for much the same reasons that being dirty causes it to run fast. Attraction between magnetized components robs power from the system, and below a certain power threshold the balance begins to oscillate in a low amplitude way, with a faster beat than it should have*. In this case, a watch can simply be "degaussed" and will return to its usual accuracy. Degaussing involved passing a varying magnetic field across the item to be degaussed such that magnetic domains in ferromagnetic metals and alloys become randomized again, instead of being aligned, which can be caused by exposure to strong external magnetic fields. It's possible to do this at home with a strong magnet, such as a fair sized neodymium magnet, but it's also possible to make things much worse by trying to do this. Degaussing in a purpose-built degaussing machine is much easier and more reliable. A watchmaker will have one of these, and can degauss our watch for you, either alone or as part of a general overhaul.
If cleaning and degaussing don't do it, then there is probably something worn somewhere in the mechanism, which is again causing drag on the whole device, robbing power and making it run fast. This is too long of a list of possibilities to go through, but suffice to say that your watchmaker may discover worn parts in the course of a regular service. Typically it takes decades of running for a good quality watch to start needing parts, especially since modern watch cases are generally very dust proof, so unless it's a vintage watch, it probably just needs a good cleaning and oiling.
* A pendulum retains the same frequency no matter what its amplitude in the absence of friction, so this isn't strictly correct. More likely is that grime adds friction/tension to the system, limiting motion and effectively shortening the length of a pendulum to increase the frequency.
I ran into this at another forum, the original content is here: http://wiki.answers.com/Q/Your_automatic_watch_watch_runs_to_fast_how_do_you_slow_it_down