The origin of the word espresso depends on what you consider its meaning to be. Some dictionaries say that it means “pressed out”, as in the coffee is pressed out from the beans using steam. If this is the case, the Italian word espresso means pressed out. The word would be Italian, and comes from the word exprimere in Latin, which means to press or squeeze out.
However, some people believe that the word espresso is related to express. In other words, the coffee is made quickly (using the steam). Or, it could be that it is a personal serving, made expressly for you. Express has counterparts in French and Italian, so those could be the roots as well.