The notation for "a quasilocal ring $(R,M)$" is shorthand for "a ring $R$ with a unique maximal ideal $M$."
It's similar to one notation used to specify a ring by elaborating on all of the data required to define it, like "$(R,+,\cdot)$".
It's worth mentioning that the usage of "quasilocal" has been declining since the 1960's and now will typically just appear as "local". The notation $(R,M)$ has persisted through the change, apparently.
In the course of writing the answer and participating in the comments, some users mentioned the possibility of using quasilocal to mean "finitely many maximal ideals."
I can say with some confidence that (in English) I have never seen that usage in a book or in an article. The term that does mean that for commutative rings, and which appears consistently is semilocal.
In a google search for "quasilocal ring" there were hits from Wolfram Mathworld and CommAlg which used the "multiple maximal ideals" version. However, I've found these two resources are not reliable. By that I mean this is not the first time I've found them to be inconsistent with the rest of algebra literature. I could not find any usages of this sort in google books. On the other hand, the Wikipedia support the usages in the way I'd expect.