A trap seals out gases and provides a bottleneck that prevents clogs from traveling farther down the line. That's why most clogs are found in a trap bend.
To clear a clog dismantle the trap and clean it out. You'll probably need to replace some rubber washers as they get brittle with age. It's not unusual to find that the entire trap needs replacing. If even one component is leaky, replace the entire trap rather than a single piece -- one worn component is likely to mean wear in others. Less expensive chrome-plated traps are notoriously short-lived. They may look OK on the outside but squeeze the bottom of the bend with your fingers to check the trap. If you feel it give, even slightly, the metal is corroded. For durability buy a plastic trap or a chrome-plated trap that is made of heavy-gauge brass.
About an hour to dismantle and reinstall a trap
Groove-joint pliers, toothbrush, hacksaw, sandpaper
Dismantling and connecting parts with washers and nuts
Prepare a comfortable work site; have a bucket or dishpan and a drop cloth in place
Washers, pipe tape, perhaps a new trap
Take the old parts to a home center or hardware store to find replacements. Make sure the new pieces will fit exactly; you may need to cut the tailpiece or drain elbow. Begin assembly by wrapping pipe tape clockwise around the threads of a tailpiece, then screw the tailpiece onto the drain body.
Assemble the pieces and hand-tighten all the nuts. Then go back and tighten the nuts with groove-joint pliers. To test the drain for leaks, stop up the sink, fill it with water, and open the stopper. Watch carefully for any leaks as the water flows through. Tighten joints where necessary.