It's often used in music to refer to a part in a composition that is essential to the overall structure and must be played. For example, a "violin obbligato" means the violin part is a crucial element of the piece and cannot be omitted.
However, "obbligato" can also be used in a more general sense to describe something that is essential or necessary for something else. For example, you might say "The cake is an obbligato for a birthday party."
The word comes from Italian, where it means "obliged" or "bound".