One, single, unified
Uni- comes from Latin "unus" (one) and signals oneness or singularity. It appears in words describing a single unit, a single form, or the act of becoming one: uniform = one standardised form; unique = one of its kind; universe = everything turning as one; unify = to bring many into one; unity = the state of being one.
Uni- vs. mono-
Both uni- (Latin) and mono- (Greek) mean "one," but they are not interchangeable. Uni- prefers formal and everyday vocabulary with Latin roots: uniform, unique, union, universal. Mono- prefers scientific and technical vocabulary with Greek roots: monologue, monopoly, monotone, monochrome. The historical root of the word determines which prefix it takes.