Cooling systems are fairly simple -- at least in the sense that your body's cooling system is simple. The system's heart -- the pump -- pushes fluid -- blood or antifreeze -- from the center of heat production to the radiator. But your pump can go bad in many of the same ways that your heart can, failing mechanically or developing internal leaks that decrease efficiency. Water pumps typically exhibit a few symptoms both before and after failure, but you'll need to take them together to correctly diagnose the pump.
Types of Failure
A water pump can fail in two basic ways. The first is that the fluid-driving impeller inside the housing fails, either because the impeller blades break off, or because the impeller has worn against the housing. With a plastic impeller, wearing against the housing will grind the impeller away; a metal impeller may grind away at the housing. Either way, internal leaks develop, reducing the impeller's efficiency. The second mode of failure involves the bearings and shaft seal. When the bearings fail, the belt will pull the on the shaft and it'll cock sideways in the housing; the shaft seal usually follows shortly afterward.
Internal Failure and Internal Leakage
Internal failures don't often cause the pump to fail outright; rather, they reduce the pump's efficiency so that it can't move the volume of fluid that it should. In the earlier stages, reduced coolant flow will cause engine temperature to run generally hotter at low speeds and briefly spike during hard acceleration. As the impeller fails further or the housing wears away, the engine will begin overheating under lighter use. Eventually, the pump will completely fail to circulate fluid, and the engine will overheat no matter how you drive or at what speed.
Bearing and Seal Failure
Water pumps aren't supposed to require regular lubrication; their bearing and seals come lubricated from the factory, and it's supposed to last for many years. But when a bearing fails and the shaft cocks sideways, the bearing will grind away and will usually emit anything from an intermittent squeak to a squeal to a grinding noise. The grinding noise is just what it sounds like; with the shaft cocked sideways, the impeller will end up sideways too, grinding against the housing and damaging either the impeller or housing. Water leaks around the shaft will follow soon thereafter, when the seal fails.
Checking Fluid Flow Visually
The old-school method of checking pump function by warming up the vehicle with the radiator cap off, allowing it to idle up to temperature and watching for water flow through the radiator still works -- provided that you have a radiator cap. Many modern vehicles fill only through the remote coolant overflow tank, so you're out of luck when it comes to checking the coolant flow visually. Unfortunately, this is the only DIY way to know for sure if you've got a bad water pump or a clogged radiator core, since they'll both exhibit the same symptoms in terms of overheating. During the visual inspection, a clogged radiator core will usually cause the radiator to overflow -- or the water level in the radiator to rapidly rise -- long before it heats up. If you don't have a cap, you can still have the radiator pressure tested to check for clogging, but it will cost you.
Watching the Gauge
When you park your car after a long drive, water will sit in the engine block and absorb heat from the metal. That's why your temperature gauge will always spike after you shut the engine down, and will quickly drop when you re-start the engine after it's had a chance to cool slightly. However, if your water pump is dead, then water won't circulate and the temperature needle won't drop as quickly when you re-start the car. So, run your car until it gets up to temperature or starts to overheat, and then park it. Turn the ignition key to the "on" position, and watch the needle. You'll probably see it rapidly rise; if not, then this alone could indicate water pump failure. Re-check the temperature after a couple of minutes, when the temperature needle just starts to fall. Start the engine; if the needle rises and the cooling fans are on, then you've got either a bad water pump or a clogged radiator. A needle that doesn't fall quickly or remains relatively stationary could indicate the same things.