Singing "in parts" means that each voice (such as soprano, tenor, alto, and bass) has its own independent line to follow. The contents of that line will be written out, and will depend on the composer or arranger and the harmonic structure of the piece. These parts may form consonances or dissonances with one another, and they may move in parallel motion (going in the same direction), contrary motion (going in opposite directions) or oblique motion (one stays on a note while the other is moving).
Singing "in unison" means that all the voices are singing the "same" line. I put "same" in quotes because, as you note in your last paragraph, they may be in different octaves. With voices, this almost always means the men are singing the line one octave below the women.