Definition of a finitely generated $k$ - algebra

An $A$-algebra $B$ is called finite if $B$ is a finitely generated $A$-module, i.e. there are elements $b_1,\dotsc,b_n \in B$ such that $B=A b_1 + \dotsc + A b_n$. It is called finitely generated / of finite type if $B$ is a finitely generated $A$-algebra, i.e. there are elements $b_1,\dotsc,b_n$ such that $B=A[b_1,\dotsc,b_n]$. Clearly every finite algebra is also a finitely generated one. The converse is not true (consider $B=A[T]$). However, there is the following important connection:

An algebra $A \to B$ is finite iff it is of finite type and integral.

For example, $\mathbb{Z}[\sqrt{2}]$ is of finite type over $\mathbb{Z}$ and integral, thus finite. In fact, $1,\sqrt{2}$ is a basis as a module. You can find the proof of the claim above in every introduction to commutative algebra.