The system of drainpipes under your sink is designed to be easy to disassemble, and replacements for the individual components aren't expensive, so you seldom have to attempt to repair one. If life with an active family keeps you too busy to get to a big hardware store with plumbing supplies, though, there is a simple fix that will stop the leaking and prevent more damage to your cabinet. Because the pipes aren't pressurized, you can patch small cracks with 2-part epoxy cement. It's better than epoxy putty, because it flows more evenly and bonds more securely.
Turn off the shut-off valves under the sink to ensure that no one can turn on the faucet while you're working. If the leak is in a vertical or horizontal section of pipe, refrain from using the sink for 12 hours to allow the pipe to dry out. If it's in the P-trap, you'll have to empty the trap.
Unscrew the compression nuts on either side of the inverted "P." If you can't do this by hand, use adjustable pliers. Disconnect the two sides of the trap and remove it. Keep it upright until you get to a place where you can turn it over and dump out the water.
Clean the hair, soap, food scraps and other gunk out of the trap with a garden hose. Empty it completely -- perhaps shaking it a few times to get out the water droplets -- and reinstall it. Let it dry for an hour or two.
Mix enough 10-minute epoxy resin and hardener to patch the crack. Spread it with a plastic putty knife, pressing it into the crack and leaving a visible amount on the surface of the pipe to cover the crack. Leave it for 30 minutes, which should be more than enough time for it to cure.
Turn on the water and test the drain. If it still leaks, repeat the procedure.