Why do all materials resonate? Why do things vibrate with resonant frequencies. 
Why are there multiple frequencies from one impulse?
ammendment:
Why do chimes from bells have overtones? How to drums have overtones? 
What is producing more than one frequency?
 A: If you make a sound, and it reflects back, you hear an echo.   If the sound echoes from multiple different places, those separate echoes are jumbled, and you hear indistinct sounds.    Similarly, the boundaries of a struck object like a bell or chime are reflectors of any sound-wave-in-the-solid,
and can produce echoes (the sound comes back to the point of origin).
Resonance means that the returned sound is not jumbled, but at or nearly
at the same phase, so there will be a second, third, fourth cycle
nearly the same as the first (because the same-phase returned sound
is similar to the original striker event).
As for 'overtones', there are resonances where a 1 cycle per second wave
returns after 1 second delay (that's called a 'fundamental' resonance),
and (given that speed of sound is same for all frequencies) there
ought to be a 2 cycle per second resonance also (because one second
delay in the echo makes for alignment of such waves after two cycles).
That would be a 'second  harmonic', and is an overtone of the
fundamental.
The striking of a drum with a soft ball generates few high harmonics, because the wide ball in contact with the drumhead makes no short-wave
displacement.  Striking the same drum with a hard stick generates higher
harmonics because the tiny tip contact footprint creates shorter waves.
Shape, material, and supports of a bell or chime can cause a variety of resonance sounds; a typical bell-tone would be three harmonics, the first, third,
and fifth, with a time-of-attack which is quick and time-of-decay
of 1 for the first harmonic, 1/3 for the third harmonic, and 1/5 for
the fifth harmonic.   Strings and drumheads and organ-pipe air columns
also have harmonics and resonances.   It can get very complicated.
Your voice isn't quite like anyone else's, because of the shape
and composition of your airways and vocal cords.   Your overtones
and harmonics define your voice to a great extent.
A: objects resonate because they contain mass and compliance. when perturbed, the mass "bounces" off the compliance and when the perturbations occur at about the same rate at which the mass in the system is bouncing on the compliance, resonance occurs. 
An object like a slender rod is fundamentally resonant at frequencies where the wavelength of the vibration equals the physical length of the rod. Since a frequency twice this will also fit within the length of the rod, that rod will also resonate at twice its fundamental frequency, four times the fundamental, and so forth. 
if the object is a thick rod, then it will resonate at wavelengths that will fit along a diagonal passing through the length of the rod at an angle in addition to wavelengths that align with the central axis of the rod. The more irregular the object is, the more different wavelengths can fit in it, and it will exhibit a whole family of resonances instead of just one. And each of those resonances can be accompanied by multiples, as explained above. 
