The easiest way is to run water into the sink (assuming it's not already filled with water), and create some extra pressure. If you have a small plunger that helps. I'd put the plunger in some water with a little bit a bleach if it's the same plunger you use in a toilet. If it's a bathroom sink, or a tub, with an overflow drain at the top you'll have to plug the overflow drain with your hand or a plastic bag while you plunge in order to create pressure in the drain.
If you don't have a plunger you can sometimes unclog a sink or tub using a plastic grocery bag as a makeshift plunger. You can also try poking down the drain with a flexible piece of wire like a coat hanger. Be careful you don't poke too hard if you have an old corroded metal drain pipe because you can poke a hole in the pipe. It sometimes works best to make a little J bend in the end of the wire and use it as a hook to pull the gunk back out the pipe.
If the clog is caused by grease sometimes pouring boiling water down the drain helps.
I've never had any luck with things you pour in a drain like Liquid Plumber or Drano. I assume anything that is so caustic even when it's diluted with water that it will dissolve hair, gunk, grease, soap etc. would be a very powerful chemical and not something stores would sell to non-professionals.
If the clog is something solid like a ball of hair you might not be able to plunge it out. You'll need to undo the drain pipe under the sink. This pipe is called a S, J or P trap based on it's shape and because it traps water in the bend so the smell of the sewer doesn't come back into your house. Sometimes undoing this pipe is really easy - if it's plastic pipe you can unscrew it with your hands. If it's an old metal pipe this can present some challenges if the pipe and fittings are corroded. It's good to have a small bucket or some towels when you undo the S or P shaped drain pipe under the sink to catch the water trapped in the pipe or sink.
If the clog isn't in the trap below a sink, or for a stubborn tub clog, you'll need to use something called a drain auger or plumbers snake. It's a long flexible metal tube with a smaller flexible wire inside. One end connects to a power drill if it's a power snake or if it's a manual snake it has a handle you can turn. You can get a Drain Auger at Lowe's for 8 bucks.
If you have a major clog in your plumbing that impacts more than one sink, toilet, tub - you'll need to find a clean-out in your main sewer (not all houses have these). It's a Y shaped connection to the main sewer that has a removable cap that allows you to stick a snake into the main sewer. If the plug is within the reach of your snake you can get it out - but if it's a tree root or something that broke the sewer pipe, or is clogging it, further away from your house you may need to call a professional...or get out a shovel or rent a small backhoe.
You'll need to get a map from your local sewer department showing where your house's sewer line connects to the city sewer and then trace from your house to the main sewer with your backhoe/shovel. Sewer pipes are cheap but they are buried pretty deep and they have to slope away from your house to the main sewer. When our main sewer got broken I hired a guy with a backhoe to dig it up for less than a 1000...the router router guy gave me an estimate of over 8000. So hiring a backhoe operator might be the way to go if you have to replace a your house's main sewer pipe.