Chemical drain cleaners may work quickly, but they are a poor choice for unclogging a drain in the long term. These solutions can slowly eat away at metal pipes, so if you use them often, you're likely to find yourself with a leak one day. They are also dangerous, sometimes causing caustic burns should they happen to splash up into eyes or face. Not to mention, using chemical drain cleaners is not great for the environment. For these reasons, it is best to avoid the use of chemical drain cleaners whenever possible, and instead rely on one of these safer drain cleaning methods. You'll need a little time and elbow grease, but one of these strategies is almost certain to clear the clog.
Pour some boiling water down the drain
It's best to try this method first thing. Either it will clear your drain, or it will loosen the clog just enough to make the other methods discussed below more effective. Boil a big pot of boiling water. Then, let it cool just below the boiling point, and pour it down the drain. If it won't all go down at once, wait for most of it to drain, and then re-heat the rest and pour it down. If the water starts flowing freely after this, you're done. If not, then move on to one of the following solutions.
Use a plunger -- the right kind of plunger
If you have tried plunging a sink drain in the past, but to no avail, your lack of success was probably because you tried to use your toilet plunger in the sink. Toilet plungers' splayed ends are designed to fit into the opening in your toilet. What you need to unclog a sink is a plunger with a plain, non-flared, bowl-shaped end.
Your chances of success will also improve with the proper method. Put enough water in the sink to cover the head of the plunger. Then, place the plunger head carefully around the drain. Plunge down with as much force as possible, and then pull up vigorously. Use a few strong, long strokes rather than fast, short ones. When you hear a loud gurgle, lift the plunger out of the sink, and watch the contents drain down in victory.
Try your wet-dry vac
If you are not able to free the clog with a plunger, this method is a good one to try since it will apply a bit more force to the clog than most people are able to apply by hand. Obtain a wet-dry vac (try borrowing one from a friend or renting from an equipment rental shop if you don't own one.) With the vacuum off, place the head around the drain opening. If it does not fit around the opening well, you can put on a rubber glove and use your hand to create an "O" shape, sealing the vacuum to the drain. Turn on the vacuum (or have a friend do so while you hold the wand.) Whatever is clogging the drain should get sucked up, clearing the drain.
Rinse with vinegar and baking soda
There are two uses for this unclogging method. You can use it on its own for a minor clog that you think is caused by grease (it won't break down hair or other tougher clogging materials), or you can use it as a follow-up to the unclogging methods above. Using it after you've removed the majority of a clog with a vacuum, plunger, or boiling water helps ensure that you've indeed removed all of the offending grime from the drain, so the clog does not just re-form.
Pour a small box of baking soda into the drain. Then, drizzle in about a cup of vinegar. Let this mixture work its magic for about an hour, and then rinse it away with warm water. The foaming action and acidity of the vinegar help loosen grease and grime.
If none of the methods above are effective in clearing your drain, then you may have a bigger blockage in a larger pipe. Your best bet is to call a plumber and have it professionally removed, since neither these methods or chemical drain cleaner will be terribly effective at loosening a clog in a larger pipe.