Begin by clearing everything out from under the kitchen sink to give yourself ample room to work. Place a large bucket directly under the P-trap (the U-shaped section of pipe) to catch any water or debris that will spill out. Lay down several old towels around the bucket to absorb any splashes and keep your cabinet floor dry. It is also wise to put on safety glasses and rubber gloves to protect yourself from dirty water and potential bacteria.

The first mechanical attempt should be with a sink plunger. If one side of your sink has a garbage disposal, plunge that side. Use a wet rag or a sink stopper to completely seal the drain opening of the other sink basin. This is crucial as it forces the pressure from the plunger down the drain instead of up into the other sink. Fill the basin you are plunging with 3-4 inches of water to allow the plunger to form a good seal. Place the plunger cup over the drain and pump it up and down vigorously for about 20-30 seconds. Pull the plunger away quickly to see if the water rushes down the drain. If not, repeat a few times.

If plunging fails, the clog is likely in the P-trap. Position your bucket directly beneath the trap. The P-trap is held in place by two large slip nuts, one connecting it to the sink's vertical tailpiece and the other connecting it to the horizontal drain arm going into the wall. Loosen these nuts by turning them counter-clockwise. You should be able to do this by hand; if they are too tight, use a pair of channel-type pliers. Once the nuts are loose, carefully wiggle the P-trap downwards and pull it free. Empty its contents (water and gunk) into the bucket. Clean the inside of the trap thoroughly with a brush and hot water in another sink or outside with a hose.

If the P-trap was not the source of the clog, the blockage is further down the drain line. Take your drain snake (or drain auger) and insert the auger end into the drain arm pipe that leads into the wall. Push the snake cable into the pipe until you feel resistance. Once you hit the clog, lock the cable clamp and begin turning the handle clockwise while applying gentle forward pressure. This will either break up the clog or allow the auger to hook onto it. You may need to push and pull the cable a few times. If you feel the snake has hooked the clog, pull it back out of the pipe slowly, bringing the debris with it. Clean the auger tip and repeat if necessary.

Once you are confident the clog is cleared, you can reassemble the plumbing. Re-attach the clean P-trap, making sure any gaskets are properly seated. Hand-tighten the slip nuts onto the threads of the tailpiece and drain arm. Ensure they are threaded correctly and not cross-threaded. Once hand-tight, use your channel-type pliers to give each nut a final quarter-turn to create a snug seal. Do not overtighten. Place a dry towel under the reassembled trap and turn on the hot water faucet in the sink. Let it run for a few minutes to flush out any remaining debris and to test your connections for leaks. Check the slip nuts carefully for any drips.