a.c. unit works when I push in relay but not when I turn on temrastat

If you have 24 volts to that contactor coil wires and it is not pulling in, it is likely open and needs to be replaced,
If you dan't have 24 volts when set to cool, may be a thermostat problem, or the next place to check.
If control wiring is exposed to outside unit, it can be damaged by people or animals.
Be sure thermostat is not just on, but set low enough to kick in.