What is timbre in sound?

Wikipedia provides a comprehensive answer here

See the “Attributes” section, especially.

This is a bit of answer to the second question, not necessarily timbre.
Attack , decay , and waveform will make the same note sound different from different instruments. Attack is the way a note rises in volume, a note from a plucked or struck instrument rises suddenly, a note from a bowed instrument can rise gradually.
Decay is the way the volume of the note decreases until it’s quiet.
The shape of the sound wave affects the sound, a nice roundish wave will sound mellow, a square wave is sort of buzz-y.

I’ll try to keep this brief and not so abstract, but I usually can’t help myself.

Timbre is determined by three things: An infinite set of frequencies, an infinite set of specific amplitudes and phases assigned to those frequencies, and the nature in which those variables change over time.

All sounds at any instant can be represented by these three variables. However, these change with time and can only be defined over a time interval if a tone is constant (such as certain simple digital synthesizer tones). The gradual or immediate variation of these variables is often associated with the word “timbre” when referring to the sound of an instrument.

So anyway, it all comes down to the sine waves that compose a particular sound. A sine wave is the fundamental sound and can form all sounds when played with other different sine waves. For example, a touch tone on a phone consists of two sine waves of two distinct frequencies added together. A saw wave contains an infinite number of them, but are limited to the harmonic frequencies and diminish in amplitude (and thus intensity) as the frequency increases.

Acoustic instruments such as pianos and guitars are much more complex with an amplitude and phase defined for every possible frequency in our range of hearing and beyond, of course changing its “shape” over time. Just as you would recognize a photograph of a familiar person or object, our ears use a similar format of information to relate certain sounds to their respective source.

Musical instruments do not play a single frequency. A large part of what distinguishes is the combination of harmonics.

Timbre refers to the quality of the sound. Like a dijeridu is all raspy and a flute might provide a pretty pure sound. The configuration of each sound maker determines all the overtones and weird sounds that get added to the basic pitch. A dijeridu has an interior that was eaten out by termites. It’s not smooth. It has all kinds of surfaces that add more complexity to a sound.

All materials have different sound profiles, and those profiles depend on the density of the material and shape of it and the way the sound is being produced. The scientists can describe it in terms of numbers, but the reality is how we hear it, and in many ways, timbre is the “voice” of the instrument.