Why is it necessary to work with different algebras?

Should one try to make all the algebraic calculations is a single universal algebraic system?

This is possible in the commutative case: just take the polynomial ring k[X₁,...,X_n] in n (commuting) variables, or in infinitely many variables, if necessary - these are the

free commutative algebras.

There is a corresponding analogue in general:

the free (non-commutative) algebras with a given number of generators. However, for most questions of interest, this object is far too large to provide reasonable answers!

For calculations with non-commuting elements, information concerning commutation behaviour is of utmost value: To codify this behaviour amounts to work in a particular algebraic system - to deal with a specific algebra.

"Algebra" is the study of such algebraic structures, the study of "algebras".