A group is a set of objects and one operation that can combine two objects in the set to make another object in the set. The operation is usually concatenation, and it must obey four rules. Rule #1 is closure; if you apply the operation to any two elements in the set, you get an element in the set. Rule #2 is associativity; a(bc) = (ab)c. Rule #3 is the existence of an identity element i; for every element a, ia = ai = a. Rule #4 is the existence of inverses; for every x in the set, there is an x' such that xx' = x'x = i. An example of a group is the rotations of a square that still fit it into the same frame. The identity is no rotation at all, the inverse of two turns clockwise is two turns counterclockwise, and so forth.
Rings are two groups that have the same set of elements where at least one of the group operations distribute over the other. If the two group operations are cancatination and "+," then distribution means a(b + c) = ab + ac. One example of a ring is the intergers with group operators multiplication and addition. Both groups obey all four group rules, and multiplication distributes over addition: a(b + c) = ab + ac. Another ring is boolean algebra, where the elements are statements that can be true or false (the inverse of X is NOT X), and the group operators are AND and OR. Both operators distribute over each other because A AND (B OR C) = (A AND B) OR (A AND C) and A OR (B AND C) = (A OR B) AND (A OR C).
Vector spaces consist of vectors and scalars. Vectors consist of multidimensional objects, and scalars are numbers. Vectors can be added and subtracted but not multiplied or divided. Vectors can also be multiplied by scalars. Examples of vector spaces are physical vectors, matrices and lists.
Quaternions are an extension of imaginary numbers. Instead of one imaginary component, i, in complex numbers, quaternions have three imaginary components: i, j and k. Similar to the way figures drawn in the complex plane can be rotated by multiplying the points by a complex number, figures drawn in three dimensions can be rotated in any direction by multiplying the points by a quaternion. This is how Computer Generated Images (CGI) are manipulated in movies.