The basic process is to push the end of the snake into the drain opening and turn the handle on the drum that contains the coiled-up snake. The auger then begins its journey down the drain. Keep pushing more snake into the drain until you feel resistance. You may have to apply pressure while cranking the handle to get it to bend around the tight curves in the trap under the sink. After clearing the curve, the snake usually slides through easily until it hits the clog.
Snakes are especially handy because they're long enough to reach clogs deep in the drainpipe.
Rotate the auger cable against the blockage until you feel it feed freely into the pipe. The rotating action enables the tip of the snake to attach to the clog and spin it away or chop it up. If the clog is a solid object, the auger head will entangle it. If you don't feel the auger break through and the twisting get easier, pull the auger out of the drain and, most likely, the offending object will come out with it. Run water full force to be sure that the drain is unclogged.