A "cut" (a term that comes from violin vocabulary) is a quick pull-off to the target note from a note that is located a half-tone or a tone above the target note. This embellishment is probably one of the most typical of the Manouche style so you really should integrate it to your vocabulary to sound Manouche. A cut can be played "binary" or ternary (swinged): it's also useful to feel the slight difference between those two.
Half-tone binary cut
Whole tone binary cut
Half-tone ternary swinged cut
Cut plus hammer
Cuts are typically used in a descending context on a unique string: if two notes are played on the same string, as soon as the second note is picked, the first one is hammered again, that is without picking, only with the left hand finger hitting the fret as on this example, and whichever the distance with the first note may be: