DIAGNOSIS That's a power loss, not a dead thermostat. The most common cause is a blown fuse on your furnace's control board or a tripped safety switch, which cuts the 24V power the thermostat needs to run.
FIX First, go check your furnace. You'll need to locate the control board (usually inside a panel on the furnace itself). Look for a little 3-5 amp fuse, often orange or yellow. Pull it out and check if the thin wire inside is broken. If it is, you need an identical replacement fuse from any auto parts or hardware store. Swap it in, put the panel back on, and your thermostat should wake right up.
IF THAT DIDN'T WORK If the fuse looks fine, the issue might be at the thermostat's wall plate. Turn off the furnace breaker for safety, pull the thermostat off its wall plate, and double-check that the small wires (especially the red 'Rc' and white 'W' or yellow 'Y') are firmly seated in their terminals and none have slipped out.
HEADS UP If a new furnace fuse blows again immediately, you've got a bigger problem—likely a short in the low-voltage wiring. That's when you call an HVAC pro, because chasing down a short in the walls is a job for the experts.