The battery is dead. Long live the battery.
If your quartz watch takes a licking but actually stopped kicking, never fear: you can easily change that battery yourself. All you need is a few tools and a new battery.
Everything You'll Need
Step 1: Know what you're working with
If your watch has a screw back, you should see several notches or a raised edge that's used in conjunction with a tool to open it. If it's a snap back, there will be a small notch or lip that's used to pop it off. If you're very, very lucky and you own something like a CWC or MWC watch (or certain Timexes), there might be a battery hatch.
Step 2: Open the case back
First, figure out if your watch has a screw back or a snap-back. If it has a screw back, follow these steps to open it. If it has a snap back, follow similar steps until you get to the point where you need to actually open the case (i.e. seat the case back holder in a vice, and seat the case in the case back holder). Now, find the little notch in your snap back case and slide the case knife in there. Apply some upward friction, and the back should pop open.
If your watch has a battery hatch (like in the below image), grab a coin and twist that sucker off.
Step 3: Choose the right battery
Now the battery should be exposed — read the serial number and order a replacement. (It's even better to determine this beforehand by looking at your watch's documentation, or, if you don't have it, looking online.) To replace the battery, pop out the old one (or use a toothpick to do so) and simply snap the new one in place. Make sure it's oriented correctly.
Step 4: Close the case back
Re-thread the case back and use the tool to close it; or snap it back on with your hand; or re-thread the battery hatch.
You should see your watch ticking happily away now. If not, you may have oriented the battery incorrectly, or you're using the wrong battery. Worst case: the watch itself is damaged. If this is the case, you may be able to order a new movement for the watch and have a watchmaker replace it — though because quartz movements are so inexpensive, it may be more cost-effective to simply buy a new watch.