My understanding is that "algebra" refers to a specific field in mathematics. Here is Wikipedia's introduction:
Algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form algebra is the study of symbols and the rules for manipulating symbols.
With an article, e.g. "an algebra", it refers to a specific set of symbols and rules, e.g. boolean algebra.
Do some fields of mathematics not use an algebra? Another way of asking it is: does all mathematics involve the study of symbols and rules for manipulating symbols? If some fields do not use algebra, can I have an example? And if all fields do use an algebra, why is it not viewed as the fundamental field upon which all other fields are based?