In a sense, engine mounts are the first line of defense in suppressing engine shakes from propagating throughout the entire vehicle.
First, we should probably talk about why engines produce vibrations in the first place. Recall that the internal combustion process is a series of explosions. They're finely controlled and choreographed, but they are forceful explosions, nonetheless. Second, internal combustion engines are a hodgepodge of pumping, whirring and rotating parts. Those parts have mass, and when put in motion, they react against the engine block with equal and opposite force. Those reactions, multiplied by thousands per minute inside a typical engine, manifest themselves as vibrations.
So, some amount of engine vibration is perfectly natural -- it's just that we don't want to feel it inside the passenger compartment. Engine mounts are typically metal flanges on the engine and along the walls of the engine bay. They're lined with thick grommets or blocks made of polyurethane or rubber to absorb some of the vibration energy from the engine when it's running.
If you start to notice excessive vibration from the engine where there was none before (and you know it isn't something else, like a misaligned suspension), a good place to check is the engine mounts. Worn engine mounts may be obvious -- often the rubber or other absorbing material will be cracked, torn, or visibly worn. Sometimes you can't tell just by a visual inspection, as the rubber may have collapsed internally. If you're not sure, see a qualified technician [source: Denby].
In addition to the engine mounts, it's a good idea to check the transmission mounts as well -- they serve the same purpose, to insulate the rest of the vehicle from heavy-duty shaking caused by the transmission.
When you call a person "stiff," it's generally not considered a compliment. But when applied to vehicles, stiffness is a good thing.