When a recorded audio signal is periodic and its period is within the range permitting audible pitch, it can be expanded into a Fourier series, a sum of sinusoids.  The amplitudes of these sinusoids determines the timbre of the signal. (Their phases will change the waveshape of the signal but not its timbre; one can’t hear the phases.)  This definition of timbre is only useful in very special situations since most signals aren’t periodic.  One can speak loosely of a time-varying timbre in case a signal is nearly periodic (its pitch or waveform are changing slowly compared to the period of repetition.)  For signals that do not have a clear pitch, sometimes the (time-varying) spectral envelope can be used to describe timbre.

Subjectively, the word timbre denotes the quality of a sound, so that the pitch, the loudness, and the timbre make a total description of a sound’s character. Of these, the pitch and loudness are somewhat understood and timbre is really a catch-all term for everything that is not pitch or loudness, that is, not understood.  In special situations, aspects of the timbre of a sound can be predicted by making measurements of the corresponding audio signal as described above.  For example, the disposition of formants in a sound’s spectral envelope can make the sound suggest an audible vowel.  That would be considered an aspect of the sound’s timbre.