They connect the engine to the chassis. In street cars, they also insulate the chassis from engine vibration.
Performance mounts are stiffer than stock mounts under the theory that more energy is transferred quicker to the drive wheels because less is taken up deflecting soft, stock engine mounts. This does work, but at the expense of comfort - stiffer mounts do increase the amount of vibration transfered from the engine to the chassis and one can feel it inside the car.
'91 Classic w/the usual stuff
"It is better to go into a corner slow and come out fast than to go
into a corner fast and come out dead" - Stirling Moss