Tonic

The tonic is the home note of a key - the most stable note in a piece, the one a melody keeps gravitating back to and the one a chord progression resolves onto when the tension lets go. Name a key, like "D major" or "B♭ minor", and you are naming its tonic plus the scale built on it. Because there are twelve notes, there are twelve possible tonics.

A key really comes down to two things: a tonic note, and the set of chords that grow out of the scale on that tonic. Knowing the tonic tells a musician which scale a song is organised around, which is often all you need to know which notes will sound good over it.

Tonic versus root

Tonic and root are easy to confuse because in the simplest case they line up: in C major the tonic is C and the home chord is C major, whose root is also C. But they are not the same thing. The root is local - it belongs to one chord or one scale. The tonic is global - it is the centre of gravity for the whole key. The G major chord in the key of C has G as its root, but C is still the tonic the music wants to return to.