An oscillation is defined as a periodic phenomenon or vibration around a point of equilibrium. There are two primary types of oscillations, but both follow the aforementioned definition. An oscillation is always periodic, meaning that it goes back and forth in some manner. However, this manner may not always be apparent. For instance, a compressional wave is technically an oscillation, but it is somewhat hard to observe its periodic motion unless one can either see it from the right angle or knows that for which to look. An easy way to check if something is an oscillation is to see if the phenomenon has an equilibrium point. In other words, when it is at rest, can be excited so that it has to oscillate in order to return to that point of equilibrium.
A driven oscillation is one in which a system vibrates and does not reduce its vibrational frequency over time. In almost every case, it requires the system to be driven. For example, a swing moving in the exact same arc every swing must be driven by either the person on the swing or another source of energy, such as a person pushing the swing. As long as the energy input to the swing remains constant, so will the oscillatory motion of the swing itself.
A damped oscillation slowly but continuously decays in its frequency over time. This means that it loses energy to the environment around it. In every-day examples, this is nearly always caused by some form of friction, which slows down the oscillation over time. These damped harmonic oscillators tend to be much more complex than driven oscillators, simply because one must always account for the dampening of harmonic motion as the system progresses in time. An example of this would be a pendulum hanging from the ceiling. Over time, the pendulum will stop, meaning that it was a damped oscillator.
It is nearly impossible to discuss oscillations without explaining waves. Waves, also oscillations, come in three forms: compressional, torsional and transverse. Compressional waves, such as sound waves, are waves that move back and forth away from and towards the primary node, which is where the wave was originally propagated. Torsional waves are simple, as they are just waves that spin, and they include things like windmills, tops, or turbines. Finally, transverse waves are waves that move up and down perpendicular to the direction in which the wave is traveling.
The process behind an oscillatory system starts with the system being disturbed from its point of equilibrium. This means that energy is put into the system. An oscillatory always wants to return to equilibrium, which means that the energy put into the system will be expended as fast as is possible for that system until the system is back in equilibrium. As the energy is being expended, the system undergoes vibratory or oscillatory motion, making it a bonafide oscillatory system. This description is analogous to most elementary school students, in principle, at least. Most young students are forced into going to school, which can be thought of as their state that is out of equilibrium. They then go as fast as their ability permits to return back to their state of mental equilibrium, which is being happy and playing with their friends outside of school.
Most people do not realize, but without oscillations, we would not be able to hear or create music or any other kind of sound. All sound waves are oscillations. On a more practical note, oscillations are used everyday to power cities, for a turbine is simply a very large oscillation, as is a windmill. There are many other oscillations in your everyday life. To find them, simply look for things that are experiencing a wave-like motion.