  The purpose of this paper is to study the structure and the algebraic
varieties of Hom-associative algebras. We give characterize multiplicative
simple Hom-associative algebras and show some examples deforming the $2\times
2$-matrix algebra to simple Hom-associative algebras. We provide a
classification of $n$-dimensional Hom-associative algebras for $n\leq3$. Then
study their derivations and compute small Hom-Type Hochschild cohomology
groups. Furthermore, we discuss their irreducible components.
