Buttonz way will work, just be careful with the bleach, the lysol works well. Look under your sink & check out how the pipes & hoses are hooked up. Most modern sinks are very simple to clean, to get at the drain pipe you might not even need tools.The drain pipe comes straight down and then turns to go back up-it'll look like a "J". That's the sink trap, it catches things that shouldn't have gone down the drain like rings, it works well, but hair & everything else that's heavy stops & stays, creating the big stink. PVC & ABS are the "modern pipes" you'll find, there will be 2 or 3 sections-1 coming down from the drain, the J and then the pipe that heads on out. Loosen the the "J" just above it & below it--keep a bucket under the pipe because it'll contain water & nasty stuff, dump it into a bucket & scrub it down-then just reverse the process--if you have an older sink you'll need a pipe wrench-2 make it easy. Either way, the first time might take an hour, after you've done it a few times it only takes a few minutes
That is typically caused by a build up of bacteria in the drain elbow (trap). Dump a bit (approx 1/2 cup) of Mr. Clean, Chlorine bleach, .. any kind of sanitizer will typically fix the problem. Let it sit for about 10 mintutes..Then fill the sink and let it drain. Filling the sink discharges water all at once which will help to exchange that standing water in the trap more effectively than just letting the tap run.