When you gesticulate, you make sweeping and excited movements with your hands when speaking. Someone describing a scary car accident might gesticulate wildly.
The verb gesticulate is related to the noun gesture. Gesture has stress on the first syllable, and gesticulate has the stress on the second syllable. Gesticulate, which comes from the Latin gesticulus meaning "to mimic," describes animated movements people make in conversation — with and without words. People who gesticulate could be said to talk with their hands!