Diagnosis
You've got a complete power failure, which almost always means the unit's safety tip-over switch is stuck in the 'off' position or the internal thermal fuse has blown.
How to Fix It
First, unplug the heater. Look at the very bottom of the unit. You'll see a small, plastic switch or button near one of the corners. That's the tip-over switch. Press it firmly with your finger until you hear a soft click. That resets it. Now, place the heater back on a completely flat, level floor. Plug it in and try the power button.
If That Didn't Work
If it's still dead, the internal thermal fuse may have blown, often from a clogged air intake. This is a more involved fix. You'd need to open the casing (voids warranty) and locate the small, glass-encased fuse on the main wiring path. It would need testing with a multimeter and replacement if it's open. For most folks, this is the point to contact Lasko support.
Heads Up
Never run this heater on thick carpet or an uneven surface. The tip-over switch plunger needs to be fully depressed by a hard, flat surface to work correctly. A wobbly floor can cause intermittent shut-offs.