Why Strings Break
Strings today are manufactured to such high-quality standards that it very uncommon for a string to break with normal use/tuning. The following is a list of common locations where strings break and the probable cause. When in doubt, keep your broken string and take the string and your instrument to your local repair person for investigation.
8 Most Common Locations for String Breakage, Their Causes, & Suggestions for Repair
1. Breaks at Fine Tuner
Most likely the sides of the tuner are pinching the string causing it to break or there is a metal burr on the fine tuner prong. Loop thick, soft-centered, synthetic core strings around fine tuners (like the one pictured above) to avoid the string becoming pinched inside of the tuner prong. 
For loop-end E Strings which attach to a Hill-Style Fine Tuner (pictured above), we recommend adding a Wittner String Protector directly to the hook/prong of the fine tuner. This tiny piece of magic plastic (pictured below) does wonders to prevent loop-end E string failures.
2. Breaks at the Tailpiece Slot
The string slots in the tailpiece are too tight for the string and are pinching the sides of the string causing the string to break. Have a repair person adjust the width of the slot for proper clearance.3. Breaks or Unravels at the Bridge
The string groove in the bridge is either too deep or too rough and the string is being pinched.
With regular use, the strings naturally kinda saw their way into the bridge over time. Bridges made out of lower-quality wood may be so soft that the strings their way into the grooves too quickly and quickly cause problems for all four strings. Have a repair person adjust or replace the bridge.
4. Breaks or Frays in the Playing Area
Ever have your teacher lecture you on your fingernail length? There's a reason! Sharp fingernails absolutely damage strings. Keep your fingernails trimmed on your left hand.
Some players who sweat more, have a more acidic response to their strings. This reaction can damage the string, causing a caustic tarnish to develop on strings. Start by making sure to wash your hands before playing, and, if you tend to wear more quickly through one specific string (like the E string on the violin), talk to your luthier about string alternatives made out of different materials.
Strings can also wear in this area from use-- some players need change their strings as often as every 6 to 8 weeks due to their unique amount of playing/practicing time. The more you play, small amounts of the metal are worn away and the string gets thinner and can start to fray or break.
Finally, an uneven fingerboard can also cause this wear. If this is the case, have a repair person check and resurface the fingerboard.
5. Breaks at the Upper Nut
Like the bridge, strings slowly saw into the upper nut with regular usage. Once the string groove becomes rough or the notch becomes too deep in the nut, the string can fray or break.
Have a repair person repair or replace the nut.
Excessive tuning from out-of-round or poorly fit pegs (they slip often) can also cause accidental breakage.
6. Breaks Between the Upper Nut and Peg
In almost every case, this breakage is caused by tuning/tightening the string too high. (Imagine pulling a rubber band too tight!)
Additionally, the transition to the upper winding of the string--where the string goes from metal to thread-wrapped--is a fairly weak part of the string. Over-tightening the string puts excess force on this already fragile place and causes the string to snap.
7. Breaks Where the String Meets the Peg
Especially common with the E and G strings on the violin or the A and C strings on the viola/cello, the string can get caught between the edge of the pegbox wall and the hole for the peg. Be sure to properly wrap the string onto the peg so that the last winding does not forcibly press against the pegbox wall.
8. Breaks in the Windings on the Peg:
If the string hits the bottom of the pegbox, it can be worn through.
Be sure to properly wind the string without numerous layers over top of one another. If the string still hits the bottom of the pegbox, then the instrument should be taken to the repair person for more space to be added under the peg for proper clearance.
Occasionally strings may hit other pegs--this often true in older instruments that have replacement pegs with larger shafts or an instrument whose peg holes were drilled in the incorrect positions. If this is the case, your peg box may need bushings to realign the pegs to prevent string contact.