There's a technique I've seen used a lot in jazz, gospel, pop singing, etc. where someone will play a note and play a quick semitone up and back down into another note. Heres an audio example. Does this technique have a name? If so, what is it called? I've heard the term "mordent" before but I don't think that's what I'm talking about.
One of the terms is "lower neighbor." This is a bit more general than terms as the lower neighbor need not be just an ornament. It's a non-chord tone played (usually a half-step) below the main tone. A mordent is the main note, a neighbor (upper or lower), and the main note again. ("Mordent" is used for one and "inverted mordent" for the other; different authors use the terms differently.)
The lower auxiliary can be used as an ornament or even be accented and be part of the melody without disturbing the underlying harmony.