The pump is the first to experience buildup. Because this is where the water flow through, and filtered. Over time, the buildup can slow down the flow of water so that it takes longer to refill and ultimately lower your water pressure.
Secondly, remove the back of the pump. This is where most fountain pumps will have a removable face. This is on one side of the pump, with some having the removable face on both sides. After removing the face from the pump, you will most likely see a propeller on one side or two.
Use a toothbrush or other type of small brush that can reach into all of the crevices in the pump to remove any sediment buildup and debris. Sometimes running hard water over the outside is enough to do the trick.
Once all these are done, place the pump back in the fountain, and all should be working. If you can, try to run clean water through the tube to flush out of the buildup.