I don't know if this is substitution, or simply saying it's the V of the V chord, it would be helpful if someone could explain this to me.
Great answers here. Here's another go with visual aids.
In Western tonal harmony, the most important chords are the I and V, the Tonic and Dominant chords, respectively. The V chord being the most dominant route to resolve back to the I chord.
Your example of the “V of the V chord” (noted V/V) is a Secondary Dominant chord, which is a borrowed chord from another key. In the key of C major, the dominant is G. In the key of G major, the dominant is D.
So, if you’re in the key of C major and play a V/V, you’re playing a D Major chord, which is borrowed from the key of G major.
They are secondary chords and they are used in analysis to indicate that a chord has a relationship to a chord in the progression that can be view from another perspective. Specifically any chord with V or V7 in the numerator like V7/V or V7/ii is used to denote a secondary dominant and V7/V will pull you to V in whatever key you are in and V7/ii will pull you to ii in whatever key you are in and both would be V7 in their own key respectively.
Most secondary chords will be secondary dominants or have very similar function, but there are such thing as non-functional secondary chords that don't quite behave this way.
I am a student composer in the Mannes School of Music.
The whole notion of chord progression as varied due to taste. In simple terms I like is the "home base" of the piece. Traveling outside the house is like exploring other chords.
V can also be referred as the "dominant" chord. In music, V "resolves" to I naturally as it is pleasing to the ear. (Take Bach's Chorale in C major as it resolves from G (V) to C major (I) -at 0:45) In that example, Bach starts in C, travels to G (V), and resolves back to C (I).
However, secondary dominant chords a bit different. Simply, we start out with the tonic chord (i.e. C major for simplicity). A dominant chord's role is to resolve back in the tonic. But, what if we want to resolve into another chord other than the tonic?
This is when a secondary dominant chord comes into play. Its function is to resolve into a chord other than the tonic. We will call the new chord we want to "arrive", or "resolve in", the target chord. We will pick Bb Major as our target chord.
To determine the secondary dominant chord, we will need to figure out the (regular) dominant of our target chord. In this case it is F major. So, F Major would be our secondary dominant chord, resulting in a sequence of C-F-Bb, or I-IV-bVII.
We are not done yet, though. Many composers like to add a 7th interval to the secondary dominant, as it allows more tension and a bigger resolution. So, our secondary dominant chord would have the notes (F-A-C-Eb).
In conclusion, a sequence with this particular example of a secondary dominant would be C-F7-Bb, or I-IV7-bVII
If this is a bit confusing, a secondary explanation may be helpful: https://music.tutsplus.com/articles/secondary-dominants-and-how-to-use-them--audio-2663
Other chords act in a similar fashion, some building up the harmony of the piece, some creating a dissonance, and some resolving to a more "pleasant" chord. For example, a very simple chord progression is I-IV-I64-V-I. The progression starts of at "home base", ventures out to IV, which strikes out to the ear. It then reaches I64 (or G-C-E if we are in C major) in a grand fashion, travels to V, which resolves back home to I.
The concept of these simple progressions is inside the big umbrella of music theory. Bach (and sometimes Mozart) give great examples of these progressions, and what chords mean and sound like.
I hope you found this helpful!