“Algebra” is a word used to describe many mathematical fields, including some fairly narrow ones and some exceptionally broad ones. Here is Merriam Webster’s definition:
Any of various systems or branches of mathematics or logic concerned with the properties and relationships of abstract entities (such as complex numbers, matrices, sets, vectors, groups, rings, or fields) manipulated in symbolic form under operations often analogous to those of arithmetic.
In middle school, the word refers to mathematics generally encompassing linear equations, graphs of linear equations, and elementary concepts for real numbers. In high school, the types of equations are expanded, and their corresponding graphs are explained. This would be a really bad list if it were meant to be exhaustive, but it’s just my short-and-sweet list.
The word “algebra” appears in the names of many graduate-level courses in mathematics, and in fact, the word appears in the names of some esoteric and head-scratching fields whose practitioners make up a low percentage of the population.
In general, algebra can refer to many things, but there seem to be some threads loosely tying these things together. Whatever those threads are – and the dictionary’s entry is a good attempt – that’s what algebra is! = )