Common words:
* Unicycle: A cycle with one wheel.
* Unicorn: A mythical creature with one horn.
* Uniform: A distinctive set of clothing worn by members of a group.
* Universe: All existing matter and space considered as a whole.
* Unique: Being the only one of its kind.
* Unify: To join together into one whole.
Less common words:
* Unilateral: Relating to or affecting one side only.
* Unanimity: Agreement by all people involved.
* Unison: Simultaneous performance of a musical part.
* Univocal: Having only one possible meaning.
* Unilateralism: The policy of acting without the consent of other nations.
* Unification: The process of bringing together different parts into one whole.
Scientific terms:
* Unipolar: Having only one pole.
* Unicellular: Consisting of only one cell.
* Univariate: Relating to or involving a single variable.
* Unimodal: Having a single peak or mode.
This is not an exhaustive list, but it gives you a good starting point for words starting with "uni-".