Structural Fatigue is the process where a compound begins to develop cracks when loaded with a time-varying load.
Ages ago (1860's) it was thought that the material was "tired" and became weak with cyclic loading. The name has stuck ever since - even though the material in the cracked area is as strong as it was new.
Most construction materials are susceptible to fatigue. It is therefore important to check this when designing permanent equipment that is expected to have a life of more than a few million load cycles (months, days or even hours for rotary machinery).
Fatigue is dependent on many factors, the main contributors to failure are:
For ferrous materials there is a lower bound to fatigue (the endurance limit), so as long as your material is under this (stress) limit, fatigue will not be a problem.
Materials like aluminium does not exhibit this property, so in theory a aluminium part will break at some point due to fatigue. Though, if the load is low enough, and the cycles infrequent, it might not happen in your lifetime.
Cracking begin in small imperfections where the material is most stressed (often near the surface, and in conjunction with indentations).
Cracks formed due to fatigue will continue to grow until the point where the remaining material cannot support the given load. At that point the material will shear off.
A decent method of spotting a fatigue failure is that the surface of the break will be filled with small ripples (striations). Each ripple signifies a change in the load - much like rings on trees tell age.
To actually spot the crack while it's in progress, one needs to design the item so that the critical crack length is longer than the shortest one you can detect. A Fracture Mechanics analysis will tell you how fast a crack will grow, and the inspection programme should be tailored to suit this.
For materials subject to slow variations in load, corrosion will add to the fatigue rate. The corrosion will fill the crack when it's open and add to the stress level when the crack closes again.
Perhaps the most notable person who have done research in this area is August Wohler - famous for his curves on Stress versus Cycles (S/N). These curves allow engineers to read the expected lifetime of their latest creation by just inputting the stress level in the structure (no wonder he's our hero!).