Every element is uniquely a sum of homogeneous elements.
Tensors are extended to an algebra by allowing the operation of abstractly adding together tensors of different types. The notion of degree, for the grading, is the same notion of degree as you should have seen defined already for tensors. The homogeneous elements are the sums of tensors of the same degree. Sometimes people grade algebras by some semigroup other than the natural numbers. The generalization to G-graded algebras where G is a semigroup is obvious enough.