The short version
Your dryer tumbles but the clothes come out cold and damp. The heat circuit is the problem, and it's separate from the motor.
On an electric dryer, check the breaker first. The motor runs on 120V and heat needs the full 240V, so a half-tripped double breaker spins the drum while killing the heat. This trips up more people than anything else.
If the breaker's fine, the usual suspects are a blown thermal fuse, a failed heating element, or (on gas models) a bad igniter. Most of these are cheap parts and a weekend-doable fix.
One rule before you start: if a safety fuse blew, the vent is probably clogged. Clear it, or you'll be doing this again next month.
What's actually going on inside
A dryer makes heat in one of two ways. Electric models run current through a coil that glows red. Gas models light a small burner. Everything else, the drum, the fan, the tumbling, is the same.
That split matters because it tells you where to look. If you've got a gas dryer, the heating element and 240V breaker don't apply. If you've got electric, the igniter and gas valve don't.
The safety chain is identical on both, though. A thermal fuse and one or two thermostats sit in the heat circuit. Any one of them failing open kills the heat entirely, even if the element or burner is perfect.
That's why 'no heat' has so many possible causes. The heat has to survive a whole series of switches, and a single break anywhere shuts it all down.
The good news: continuity testing marches right through that chain. You check each part in order and find the open one. No guessing, no throwing parts at it.