I am just jumping into this for fun. I think they all capture the idea of a big idea or pretty big idea, but feel free to argue.
variable - we can work with quantities, and learn things about them and draw conclusions from them, even when we don't know what they are.
inverse operations - operations can be undone (and sometimes they can't, at least not uniquely) and this is useful for solving all kinds of problems.
functions - Different kinds of rules that map a set of numbers to another set of numbers follow certain patterns.
transformations - rules can be changed in systematic and useful ways.
equivalence - How do we know when things are the same? How do we know when they are not the same?
proof - Usually a big feature of a geometry class, but I'd argue at least as important in algebra. How do we know for sure that something is true?