Meaning "two" or "twice":
* dichotomy: A division or contrast between two things that are entirely different
* dilemma: A situation where a difficult choice has to be made between two or more alternatives
* diphthong: A sound formed by combining two vowels in a single syllable
* dinosaur: A large extinct reptile of the Mesozoic era
* diptych: A painting, photograph, or other artwork consisting of two sections hinged together.
* disaccharide: A sugar composed of two monosaccharide units linked together.
Meaning "apart" or "away":
* digress: To stray from the main subject in speaking or writing
* dilate: To become wider or larger
* diffuse: To spread out or scatter
* diverge: To move in different directions
* divorce: The legal dissolution of a marriage
Other meanings:
* diurnal: Occurring or active during the day
* diminutive: Very small
* direct: Going straight to a particular destination
* disrupt: To interrupt or break the smooth progress of something
* district: A region or area
* diverse: Showing a great deal of variety; very different
This is just a small sample, there are many more words with the prefix "di-"!