Tritone substitutions are a common harmonic device used in the jazz idiom. They are most often found in ii-V-I progressions, with the tritone substitution being used in place of the V chord. For example: In a standard ii-V-I progression in the key of C major, the chords would be Dmin7 - G7- CMaj7
If we use a tritone substitution the chords become Dmin7- Db7- CMaj7. (Db7 is the tritone sub for G7)
In these examples we are only looking at the Root, 3rd and 7th of the chord because these are the most important chord tones. Notice in the example below that G7 and Db7 share 2 of the 3 chord tones. The 3rd of G7 (b) becomes the 7th of Db7 and the 7th of G7 (f) becomes the 3rd of Db7.
The only note that is changing is the root, and that note is moving the distance/interval of the tritone (also called an augmented 4th/diminished 5th). This is why it's called a "tritone" substitution. It works because both chords have the same function, to move to resolution at Cmaj7. The tritone substitution adds a bit more tension to the progression and creates a descending chromatic bass line through the ii-V-I (D-Db-C).