What pieces are prone to breaking?
Lego is generally quite robust, but there are potential weaknesses:

Material Problems

Some parts are prone to damage as a consequence of the material from which they are made:

Plastic


Sunlight can cause discoloration
Bricks can be damaged by extremes of temperature
Older bricks (pre ABS) are prone to warping, discoloration and may become brittle
Most plastic bricks are easily damaged by pressure (eg. from teeth) and abrasion


Stickers


Most are easily damaged by water
Wear easily
Are easily damaged when they span components


Rubber


Rubber bands tend to perish


Structural Problems

Damage can also be caused due to the shape and structure of some components:

Long Parts

Long thin parts are prone to bending or snapping under load, for example:


Long technic beams
Flags, signs etc. with long stems
Larger plates and baseboards
Swords
Propeller blades


Clips and Other Joints


Lego clips are prone to breaking under stress, for example:


Minifigure hands
Flags that clip on to flag poles
Bricks with clips on
etc.

For older types of window frame, the lugs that hold the shutters can snap off quite easily:





Ball sockets can break if you force the ball out by levering it against the socket instead of pulling it straight out.





The hinge holes on chests can be stripped out by pushing the lid open too far




Technic Parts


Wheels and gears can crack under stress
Smaller items that fit on technic (axles bushes and the smaller cogs) are easily crushed




Other Parts


It is rather easy to supply excessive force to angle brackets eg. by stepping on them or when they are at the junction long parts that can apply leverage


Electronics

Electronic items have special problems:


Battery boxes can be ruined by leaking batteries
Electrical items can be damaged by applying the wrong voltage, eg. 12v to 4.5v lamps
Most electrical items can be damaged by water
Cables break easily, and (at least in the case of the old 4.5v system) can easily be pulled from plugs
Some conductors are prone to corrosion


Recommendations

I recommend:


Sort lego before washing to remove parts that will be damaged by water
Don't play with lego on the floor where it can be stepped on
Keep bricks out of sunlight
Don't let bricks get too hot
Keeping lego away from children(!) and pets
Remove batteries when not in use