Well, if your fish tank isn't pumping water, there are a couple of things you want to look for. A lot of times, it's the most obvious thing. Maybe it got unplugged. Maybe the surge strip is unplugged. Sometimes, when people do maintenance, they accidentally plug their filter into their light timer. So the filter could be on the same timer as the light, and it could be off. So look for the obvious things first.
Then you want to make sure that it's not clogged. A lot of times, if people neglect their filter systems, it gets so clogged to the point that the water doesn't even flow through it. So a clogged mechanical filter or a blocked passageway from a carbon filter bag that hasn't been cleaned, and now it's got a solid film of muck or bacterial slime on it, that will impede the water flow, preventing your filter from working.
So first, look at the obvious and make sure it's plugged in. You can also feel it and see if it's warm. If it's warm, that means that it is plugged in and the motor is probably either just in a neutral state. So you want to unplug it, let it cool down. If you're handy, you can take it apart to the point that you feel comfortable with. Clean the impeller. A lot of times, on overflow-style or hang-on-style filters, the impeller gets a piece of hair in it -- human hair -- or just something that gets clogged around the impeller shaft. So taking the impeller out, cleaning it with a paper towel, making sure nothing's clogged or nothing's stuck on it, and putting it back in will often free up the filter and get it to work again. Sometimes the impeller gets dislodged. You have to take it out and just reset it.
Another thing to look for is evaporation in the aquarium. Sometimes your water level evaporates to the point where a big air bubble gets stuck into the intake tube, causing an air lock in the filter. Canister filters are very prone to this. Sometimes the air stone will move itself underneath the intake of a filter, and air goes into the filter, sits at the top of the canister filter, and you have an air lock and the canister filter doesn't work. So you can take the canister filter, open it up, fill it up with water, get it primed again. That should solve it. Otherwise you can take the filter, turn it on its side very slowly to get the air bubble to shift away from the head where the impeller is. Once the filter kicks in, you can hear it start to push water again, and then slowly invert it back so that the air bubble slowly gets grabbed by the impeller and gets pushed out back into the aquarium.
But a lot of times, it's either that the filter is clogged from not being cleaned properly, or it's unplugged, or it's got an air lock. Those are the most common reasons why a filter won't flow water.