First thing to do is check the pressure in the tank. While doing that, you'll also verify that the bladder in the tank is not shot.
After the pump has run, if you knock on the tank does it sound like it is entirely full of water? If so, it is probably shot and needs to be replaced. With a tire pressure gauge, check the pressure. If the pump just turned off, that should be the high pressure limit. A pressure switch is usually something like 20/40 or 30/50, but could be anything near this. The lower number is when the pump should turn on, and the upper pressure is where it turns off. If you get any water out of the tank when you check the pressure, it is shot, replace it.
If that's okay, then switch the pump power off. Open a valve somewhere and let it drain all of the water stored in the tank out. Then, measure the pressure in the tank again. This should be 2 pounds LESS than the lower number where the pump turns on. If not, use something like a bicycle pump and pump it up. You can do it with a compressor, but on a small tank, be careful not to overinflate it.
Ideally, you have a pressure gauge on the water system so you can see what's going on. The pressure should smoothly go from the high point back down to the low point as water is used, then the pump would turn back on to try to maintain or increase the pressure back up to the high point. Depending on the pump, the amount of water in the well and the recovery rate, you might be using it faster than it can pump it so it momentarily runs dry. Then, the only thing to do is either store more water and pump slower, or build a reservoir, or drive a new well. If you are in a drought area, it is very possible you're running out of water.