Unfortunately terms are a bit overloaded. When some people say algebra they might mean algebra over a field. A field alone adds a lot of structure to your set and is powerful enough to define vectors on. Algebra over a field adds even more.

But that is different from an algebraic structure. An algebraic structure is just some set with some operations and laws. It can be magmas, monoids, groups etc.

And abstract algebra is also a necessary term to differentiate from "ordinary" algebra that is taught in middle schools and high schools, where algebra only means using letters to substitute numbers.

It's unfortunate.