"Obbligato" is a musical term that used by composers - the creator of the work. It indicates that the certain element or part is extremely important to the music itself, and should not be omitted or altered.
In music or other art forms, works are often re-arranged, simplified, or even omitted by other musicians (performers or composers) or artists. Therefore, to maintain the integrity of the piece, composers use the term "obbligato" to indicate that the certain part is "indispensable".
The opposite of "obbligato" is "ad libitum", a notation that indicates that the related parts can be changed or omitted according to the needs of the performer or conductor.