  All subalgebras, idempotents, left(right) ideals and left quasi-units of
two-dimensional algebras are described. Classification of algebras with given
number of subalgebras, left(right) ideals are provided. In particular, a list
of isomorphism classes of all simple two-dimensional algebras is given. In the
study of ideals and subalgebras the number of them depend on roots of certain
system of polynomials at structure constants of the algebra. We also give
explicit forms of the polynomials.
