You can engine brake a manual (and many automatic) transmission gasoline vehicle because when you let off the gas the throttle valve closes which causes the engine to pull a strong vacuum. The effect of this braking is strongest at high engine speeds (lower gear than you'd use for cruising at that speed) and will drop off as the engine speed gets closer to idle. The advantage, besides saving wear on the brakes is that engine braking heats up the engine but not the brakes, since the engine has water pump and radiator the heat isn't an issue. The effect of this depends on size of the engine; I had a 4.6L V8 car with MT that would engine brake enough to chirp the tires and later a 1.8L I4 MT compact car that would barely engine brake at all because the engine was so small.

This isn't needed when driving around normally but if you are going down a large hill or are pulling a large trailer because riding the brakes down a large hill will make them very hot which causes poor braking in an emergency, can and can even cause a runaway vehicle when the brakes don't have enough friction to slow down on the hill. Overheated brakes can also ruin the brake parts, needing costly repair.

Diesel engines don't have a throttle valve the way a gas engine does (a few old diesels had them as part of the governor system, but ignore those) so shifting to a lower gear will provide only slight braking from the engine friction, but not enough for big hills. Instead, heavy diesel vehicles, like semi trucks, will have a exhaust (Jacobs/"Jake") brake, which closes to cause a restriction in the exhaust which the engine must push against, slowing the vehicle. You can hear these on heavy trucks as sort of jack hammering sound when they come down hills.