The term “Augmentation” can refer to several different concepts in the realm of music.
Rhythmically, augmentation refers to prolonging the lengths of notes in a particular series of note onsets. Essentially an augmented melody is a melody expressed in longer note values, which are generally proportional in a very specific ratio to the original melody (e.g. 2:1 – quarter notes become half notes, etc.).
Diminution refers to the opposite of augmentation: when a melody is re-expressed in shorter values, proportional to the original melody’s durations in a specific ratio.