A formant is a frequency that an instrument or a voice always produces independently of what note the instrument is playing.
Woodwind instruments, for example, have formants due to their design. They resonate at a specific frequency because of their construction. A flute, for example, is expected to produce the frequency around 800 Hz (somewhere between a G and G#). Instruments can have more than one formant and the formant does not have to be an exact frequency (i.e., two clarinets may vary in some range).
Formants are important as their presence determines the timbre (the qualities) of the sound produced by an instrument. In an artificial digital pitch shift of a recording, for example, if the pitch shift does not take into consideration the formats of the sound produced by an instrument, it will change the instrument’s timbre even if pitch shifting correctly.