Without a doubt, the most common failure mode I see in my lab is fatigue. Unlike overload failures, which occur more or less immediately when a load is applied, fatigue failures are progressive in nature. In other words, they develop over time as a result of repeated loads. If you've ever bent a paperclip back and forth until it snapped, you've seen fatigue at work.
The word fatigue implies that the material is somehow getting tired, or that its properties are becoming exhausted by repeated application of loads. Of course, this is not actually the case. If you want to convince yourself of this, you can cut a small sample from of a part that failed in fatigue, and perform mechanical tests on the sample. In almost all cases, you will find that the strength is the same as it was before the part was put into service. This demonstrates that fatigue is not a matter of the material losing its strength or degrading over time. So why does fatigue occur?
Fatigue failures start with the formation of a small crack, which grows with repeated loads. Once the crack gets big enough, the part breaks.
Crack growth can be understood as an energy balance. When you apply stress to a part, you increase its strain energy. You can think of this as the energy stored in a stretched spring. Now let's think about a part with a crack in it. The surface of the crack has a surface energy associated with it. Making the crack larger increases its surface energy. So a part that has a crack in it can lower its strain energy by increasing the size of the crack by converting strain energy into surface energy. This is why a crack can grow under repeated loads. (Incidentally, it also explains why you can cut glass by scribing a line in it with a glass cutter, then bending it.)
But what causes the initial small crack to form, especially if the overall stress is low? Most parts contain stress risers, where the local stress may be significantly higher than the overall stress. Cracks initiate at these stress concentrations.
Some stress risers are geometric in nature, and can be minimized by good design. These include sharp corners, notches, and holes, among others. Other stress risers may come from manufacturing processes, and can be minimized by good manufacturing practices. These include casting porosity and other kinds of defects. Still, other stress risers may be internal to the material itself. For example, most steels contain nonmetallic inclusions, which act as internal stress risers. Bearing quality and aircraft quality steels are produced using special steelmaking techniques to minimize their inclusion content.