If your favorite sweater has sprouted a big hole, you should take it to a professional for reweaving. But if it's got nothing more than a small snag or unraveling thread, you can keep the sweater from unraveling further with this simple repair.
Insert a very small crochet hook (about a size 5) from the back side of the sweater through to the front at the site of the snag.
Use the hook to pull the loose yarn through to the back side of the garment.
Turn the sweater inside out.
Use the crochet hook to make a loop out of the loose yarn, then pull the loose end through the loop, creating a knot on the inside of the sweater.
Dab the end of the loose yarn with liquid ravel preventer (sold under the brand name Fray Check), available at fabric and hobby stores. In a pinch you can use a small dab of clear nail polish instead.
After the ravel preventer has dried, turn the sweater right-side out and gently stretch it to smooth out any bunching that the snag might have caused.
If you don't have a crochet hook handy, you can probably use a needle, a long, thin letter opener, or even the cap of a pen to pull the loose yarn through the sweater. Just be careful not to snag more threads or poke another hole in the sweater.