For those who may not have given it much thought, yes your battery driven watch is most likely a quartz watch. They utilize a quartz crystal as an oscillator, and can keep very good time. The cheap ones will drift up to 140 seconds per month, the really good ones will be within 5 seconds per month, and the Precisionist Quartz by Bulova is good for 8 seconds per year.
The heart of the watch is the quartz movement, and depending on the watch, they can be repaired or replaced, but most often they are replaced.
If your watch is a good quality one, it may be possible to simply replace the movement, and this is what happens at a repair depot usually. The older quartz watches have robust movements that are obsolete, so sometimes these can be repaired, but with parts being restricted by companies like Swatch, (Swatch owns Longines, Rado, Omega, Tissot, Breguet, Certina, Mido, Hamilton, Calvin Klein, BlancPain and ETA, the movt maker) this may no longer be possible in most cases. This also applies to almost all modern replacements of Swiss quartz movements.
Japanese movements, depending on the maker, are still widely available and there is no move to restrict them in the near future. Chinese movements can be quite inexpensive to replace and are almost never repaired.
Life span for a good quartz watch is about 35 years, for a mediocre one it can be less than 5.
If your quartz watch is kaput, it may be because of a variety of factors. Leaking battery, magnetic fields, rough handling (extremely common) dirt, wear and tear, and water damage. Sometimes they also just wear out.
As a repair shop, we either repair the movt, or replace it, the latter being most common. Quartz movts can range in price from a few dollars to hundreds or even thousands.
The long and the short of this story is Quartz watches do not last as long as mechanical watches, and they are easily destroyed by contamination and magnetic fields. We see a lot of watches with fingerprints burned into the batteries or on the movts themselves. It is important to not touch the battery or the movt with your fingers. If you see someone touching your battery or your watch movt with bare fingers, this is wrong and will cause damage.
Those ten and twenty dollar watches are basically throw away items and we generally do not recommend repairing them.