When we, for example, pluck a string on a musical instrument, it does not just return to where it was, but it vibrates back and forth around the place where it was before it comes to rest at the place where it was. What causes this visible vibration? Is resonance another name for this, or is resonance the audible counterpart of vibration?

When you pluck on the string, the contact places a force on it, which moves it away from its equilibrium (at rest) position. When the force is removed, the string, now slightly stretched, moves back toward its original position, but still has speed when it reaches it, and moves past. The tensile stresses and elasticity of the string pull it back again and past the rest point, and the cycle (vibrations) continue until they eventually dampen back to the original position. When the string vibrates in this manner, it does so with a certain 'natural frequency', a certain number of cycles per second. This frequency is sometimes called the 'resonance' frequency, and the value depends on a number of factors, including the initial string tension, length, diameter, density, and elastic properties. This frequency determines the pitch of the sound which results from the vibration. Now if you were to vibrate the string by applying a periodic motion to it that matched the natural frrequency of the string, the string would start to 'resonate' with high amplitude vibrations that could break it. This is how a singer can break a wine glass...the pitch (frequency) of his/her voice match the natural frequency of the glass, causing it to vibrate excessively and break. Natural frequency is very important in earthquake design, the buildings must be designed to have natural frequencies that do not match the frequency of the ground movements caused by the quake, else they will vibrate with large deflections to failure.

Well, I learned that one string can have all those qualities and that a wine glass, as still as can be, also has a frequency.

Just a curiosity in passing : some French makers of double-glazing panels for windows have a neat trick. They make one of the two sheets of glass of a different thickness to the other. This is to improve the noise protection. A noise outside may have a main component which is of the same resonant frequency as one sheet. So, that sheet resonates and the noise passes on;but the second sheet will not resonate with it so the sound is not passed on so readily into the room.