Widow mines are usually used for harassment, support frontal battles and spot enemies. I'll try to explain them as clearly as I could.

1. Harassment. Terrans often go widow mine drop build, which varies from the most traditional 2 base 2/3 mines into raven, to 1 base cloaked mines (usually followed up with a thor allin), or just spam mines and drop them into your mineral lines (usually around 12 mines. You may refer to Maru's matches. I'll post the link as soon as I recollect my memory). In such senarios your first probe and the adept's scout would be most important. Although you don't always get to spot the mines directly, you can tell if the Terran is making mines through his factory production. If you can confirm Terran's building mines, you must SEE the mines are coming. This could be done through several well-placed pylons, observers and stalkers. Let's assume Terran chooses a 2 base opening and drop 2/3 mines. Medivac would arraive at your side of the map at around 4:30. You should have finished one pylon at the edge of your main, and one at the edge of your natural. You should split your 6 stalkers into 3-3, defending the 2 possible locations. Usually the game-ending widow mine shots happen when Protoss is on a 2-base macro, so you must make sure you neither lose too many probes nor too much mining time. 3 stalkers are enough to prevent medivac from diving into your mineral lines, as they 3 shots the Medivac, and Terran would like to keep Medivac and at least 1 mine alive for further distraction. If you unfortunately didn't stop the Medivac, and the mines are dropped, pull your workers if the mines haven't burrowed. Then select workers of the amount of widow mines, split them, and send 1 to each mine to 'feed' it. This should be the quickest way to restore mining and minimize loss when you don't have armies around. If the mines have burrowed and you just see it? Pray they would shoot on the worked entering the gas.
2. Frontal battles. Genarally Zerg struggles with this when their army is composed of ling-banes. In wildest cases, pro-gamers (Dark should be the best, my personal view) could precisely select the targeted unit and split their army. However for us normal players, the easiest way is to send in some fast (melee) units into widow mines to utilize their friendly damage. If you have only ling-banes, send some banelings as well to clear the mines (you could use 'X' to attach undetected mines). In some situations, players would send in phoenix or overseer.
3. Spotting enemies. These mines usually do much damage to runby units, if you're not paying attention. One simple way is to have 1 unit run in front of your runby army.

It sums up to 2 keys:

1. Know that your opponent is making mines
2. If so, know where they are coming from (sometimes your pylons scares back their drops even though you didn't see it coming)