to see if it is pumping coolant
Take off your radiator cap, start the motor. If the water pump is working you will see the water flowing in the radiator. Do you have a temperature guage on your dash or just a trouble light. If you drive it and it does not overheat then it is working.
If your radiator has a cap you can take it off when engine is cold. Start the car and notice the gently flow of coolant and then after the engine warms and the thermostat opens, about 10 minutes at idle, the upper hose should start feeling warm and the coolant should flow faster past the radiator opening. If you don't see the rushing coolant the impeller of the water pump isn't working. Hope you don't have a leak but you can lose the bearing in the water pump without losing the seal, so there is no leak.
you can leave the radiator cap off and see if it circulates or you can pull 1 of the heater hoses off and see if anti-freeze is coming out.
It is working if you're not overheating, both radiator hoses get hot, and you see lots of coolant movement in the radiator after warm up.
On some vehicles the waterpump impeller(on the backside of the waterpump) may become detached or rust away. Since you are not being vehicle specific, just taking the belt off and spinning the waterpump pulley may not show you whats wrong. A thermostat that is stuck closed can restrict circulation as well.
