An algebra, as far as I know, is closely related to a group with a family of functions being closed under addition, scalar multiplication and then the product of any two functions in the family.
Then there is this separate term I came across on Wikipedia called a Lie Algebra which doesn't look related. But is there a relation? Does a Lie algebra need to satisfy the regular algebra axioms?
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Let me add that a pretty useful example is the vector space $\mathbb R^3$ over the field $\mathbb R$ where the extra operation is the cross product (also know as the vector product) which usefulness lays in the great number of geometrical constructions as well as of the vector-calculus processes.
Abstractly, both an algebra and a Lie algebra are vector spaces equipped with a multiplication. The difference is in the kind of multiplication they are equipped with. An algebra is often called by it's full name, an associative algebra, since the multiplication is associative: $$a(bc) = (ab)c\,.$$ For a Lie algebra, the multiplication has to be skew-symmetric and follow the Jacobi identity, which respectively mean: $$ xy=-yx \qquad x(yz)+z(xy)+y(zx)=0$$
Then, based on what you said at first, it might be helpful to mention that sometimes folks use the word algebra much more generally to mean "a ring acting centrally on a ring" or something similar to that.