I know that there are resonant frequencies in electronics as well as vibrational resonance. Are these related at all? And, while I can hear things resonate with sound, what exactly does circuit resonance mean? How is it manifest?
An electronic circuit that includes reactive components (inductors, capacitors etc) have a resonant frequency where the inductive reactance and the capactative reactance are equal and cancel each other out. A series resonant circuit will have minimum impedance at that point, and thus maximum current flow at the resonant frequency. This is used to pass frequencies of interest, as in a radio transmitter or receiver, and to attenuate all other frequencies. A parallel resonant circuit has maximum Impedance at the resonant frequency, and so can be used to block out specific frequencies or bands of frequencies.
Vibrational resonance is somewhat similar, in that the mechancal response or vibration tends to be maximum at the frequency of vibration as the vibration oscilates in phase with the natural movement of the structure. Bridges have famously collapsed when the wind induced a vibration at the bridges natural harmonic frequency, causing the vibrational amplitude to grow and grow to the point of dramatic failure.
A fundamental similarity between mechanical vibrational resonance and electrical circuit resonance is that in both cases the "resonance" comes from an exchange of energy between two (or more) linked modes of storage. In the case of a mechanical vibration, energy storage is commonly in forms like elastic strain and kinetic energy. In a simple mass-and-spring resonator, energy is exchanged between the spring strain and the mass kinetic energy. In electronic circuits, energy storage is commonly in forms like electric field and magnetic field (as in an inductor-capacitor resonant circuit as described by MetMan).
It's possible to have cases like piezoelectric resonators, where "mechanical" and "electrical" energy transfer mechanisms are tightly linked, and we can, for example, use a mechanical resonator, like a quartz crystal, as an electronic signal filter.