Tonic - the "home note" of a key. For example, the tonic of G major is G. The tonic of G minor is also G.
Dominant - The 5th note of a Major or Minor scale, as well as the major chord built on that note.
Leading Tone - The seventh note of a major scale, which is one half step below the tonic. In G major the leading tone is F#. In G natural minor there is no F# in the key signature (2 flats), so the F natural is raised to an F# in order to create the tension of a leading tone.
Scale Degree - Each note of a major scale can be assigned a number, counting up from the tonic (which is 1). For example, in the key of G the note B would be the 3rd scale degree, or note in the scale. Every scale degree has a unique name, but numbers are simpler and work just as well.
Nashville Numbers - Numbered chords as they relate to the home key. Just the same way we can number the scale degrees we can assign a number to the chord built on each scale degree. For example, the IV chord in G Major would be C Major, which is the chord built on the 4th note of the scale.
Diatonic - The group of notes and chords belonging to a single key. All the natural notes A through G are diatonic to the key of C Major.
Chromatic - Any note or chord that does not belong in a key (an accidental is a big hint). Here we have ventured outside of the diatonic realm.
Interval - The distance from one note to another.
Mode - A variation of the Major scale, where the tonic (home note) has been shifted to a different note. For example, if we play the notes in C Major but start and end on D instead, we just played the 2nd mode or variation, starting on the 2nd note. This is the Dorian mode. It sounds minor. The key signature is the same as C Major, but D is the home note.
Tag - A lick or statement placed at the end of a phrase, solo, or section of a tune. Most often a restatement of the last lick before ending the song.
Cadence - The end of a musical phrase or section. Usually the music will arrive back to the 'home' chord, or alternately pause or rest on the dominant.
Relative major and minor keys - The Major and Minor keys that share a key signature. For example, E minor and G Major both have the key signature of one sharp, therefore they are relatives. This applies to pentatonic scales as well, which we will see later on!
Tempered intonation - equivalent intonation of 1/2 steps, or modern tuning of pianos and fretted instruments. This allows these instruments to play in any key at any time and be able to maintain relative intonation. Mathematically however, the difference in vibration of these 1/2 steps is an irrational number!
Just intonation - intonation based on rational whole number fractions. A 'warm 3rd' in this case is a little flatter than a 3rd based on equal 1/2 steps, or tempered intonation, and as a vibration can be expressed in the fraction 5/4.
This is a tuned C# against an A drone.
A warm 3rd will be "flat" according to the tuner!
This is a G# (tuner says Ab) against an A drone
And a leading tone can be tasty a little "sharp"
See this article for more on the physics of tuning.