An Observer, by simple terms, is someone that assists in seeing all the possibilities that occur in the world of BlazBlue due to all the time loops that happen because of the Master Unit, Amaterasu.
It is unknown how one actually becomes an observer, but it is implied that one can be given the position by Takamagahara by having some degree of power, as the known beings that are observers are the series most powerful characters story-wise, or by becoming an "uncertain factor" in the time loop. It is also shown in Hakumen's story that someone can actually agree to be observed by another individual. Unfortunately, becoming an observer means one must also abide by the title and cannot take action in most situations, which is a trait that has greatly annoyed Rachel.
The idea of an observer is based on quantum physics and the Schrödinger's cat experiment. Special beings like Hazama, Hakumen or Rachel will disappear without an observer, though apparently Ragna's hatred for Hazama is enough to maintain him.