Unlike a toilet plunger, which is a virtual necessity in every household, a sink plunger is optional. It can clear clogs, but if you're resourceful, you can find a number of things around the house that can work as well or better. None of those things should include caustic soda, the main ingredient in most commercial drain cleaners. It's bad for your pipes, the environment and for anyone who disassembles the drain to clean it.
Many drain clogs are the result of hardened grease or soap in the drain tailpiece (the straight length of pipe directly below the sink) or in the P-trap (the U-shaped pipe below the tailpiece), and the easiest way to dislodge either is to melt it under a stream of hot water. Try turning on the hot water faucet and letting it run for a minute or two. Keep an eye on the sink so it doesn't overflow. If you need even hotter water, boil 1 or 2 quarts on the stove and pour the water directly into the drain. There's a good chance that this will clear a slow-moving drain, but it's unlikely to have an effect on a stopped one.
Hair can cause clogs in bathroom sinks, while stringy waste from vegetables such as celery and corn can do the same in the kitchen drain. Hot water won't dissolve these, but the foaming action created by mixing baking soda and vinegar might. Pour about 1/2 cup of baking soda into the drain and follow this with 1/2 cup of vinegar. The combination begins foaming immediately, and if you put in the drain stopper to confine the foam to the drain, it can dislodge hair, vegetable fibers and other gunk. Even if it doesn't remove the clog, the foam cleans and deodorizes the pipes.
When a plunger isn't available, plumbers often reach for a sink auger, also called a snake. If you don't have one of these, you can make a facsimile out of a wire coat hanger. Cut a straight length of the wire and use pliers to bend one end into a small hook. Remove the stopper from the drain opening. If it doesn't pull right out, locate the stopper rod, a metal rod extending into the tailpiece under the sink. Loosen the nut around the stopper rod and slide the rod out of the drain, freeing up the stopper. Feed the hanger into the drain and use it to fish out hair and other gunk from the tailpiece. Replace the stopper and retighten the stopper rod nut before flushing the drain with water.
Disassembling the P-trap is the best strategy when you can't dissolve a clog with liquid solutions and the blockage is too far inside the drain to retrieve with a coat hanger. Place a bucket under the trap. If the trap has a cleanout plug at the bottom of the bend, unscrew the plug by hand or with pliers, then use a wire or other tool to extract the clog through the cleanout hole. If there's no cleanout, unscrew the connectors on both ends of the trap and remove it from the tailpiece and drain pipe. Once it's off, you can wash the trap out with a hose and use the coat hanger to remove obstructions in the drain and the tailpiece. Replace the trap and tighten the connectors by hand.