Suppose that $R:=k[x_0,\dots,x_n]$ and $I$ is an ideal. Is there any relation between finding the minimal generators of $I$ and the graded betti numbers of the module $R/I$?

To support J.C. Ottem's answer, let me present one example. Let $R = \mathbb{C}[x,y]$ and $I = (x,y^2)R$. What is the minimal graded free resolution of $R/I$, equivalently $I$? That is, $0 \rightarrow R(3) \stackrel{d_1}{\rightarrow} R(1) \oplus R(2) \stackrel{d_0}\rightarrow R \rightarrow R/I \rightarrow 0 $ where $d_1 = (y^2 \;\; x)$ and $d_0 = (x \;\; y^2)$. Now, ask what are the graded Betti numbers and minimal number of generators for $R/I$. I agree with J.C. Ottem's opinion on reviewing the definitions. I hope this helps. 

