The definition of a finitely generated $K$-algebra is if $R$ is generated as a ring by $K$ together with some finite set $r_{1},...,r_{n}$ of elements of $R$. I want to know what does that mean?
In fact I also want to know what does mean by generated as a ring by $K$?
On
This simply means the elements of $R$ are polynomials in $r_1,,\dots, r_n$, i.e. the $K$-algebra homomorphism: \begin{align} K[X1, \dots,X_n]&\longrightarrow R\\X_1&\longmapsto r_1\\[-2ex] \vdots\;\\[-0.5ex] X_n&\longmapsto r_n, \end{align} is surjective. We usually write this as $\;R=K[r_1,\dots, r_n]$.
R is k-algebra means there is a ring homomorphism f from k to R .R is finitely generated means there are x1,x2,..,xn in R
such that any element is polynomial of x1,x2,..xn with coefficients from f(k)i.e. range of f. Any finitely generated k-algebra is isomorphic to quotient of k[x1,x2,..,xn] for some n.