DIAGNOSIS The thermostat is sending the correct signal, but the HVAC equipment is not responding. This is usually a system control issue, not a thermostat fault.
FIX Go to your furnace or air handler. Listen for a click from the control board when the thermostat calls for heat or cool. If you hear a click but the blower or burner doesn't start, the problem is in the equipment. Check the furnace door safety switch, flame sensor (for heat), or capacitor (for AC condenser).
IF THAT DIDN'T WORK The thermostat may have a loose wire connection. Power down the system and check that each wire is secured under its correct terminal screw on the wall plate. A loose Y (cool) or W (heat) wire will prevent the signal from getting through.
HEADS UP Do not assume the thermostat is broken. 'Equipment running' means it's completing the circuit. The fault is almost always downstream in the HVAC unit itself.