These concepts are related. Let $A = M - N$ and consider the iteration
$$
M x_{k+1} = N x_k + b.
$$
We can write this as a mapping $\Phi : \mathbb{R}^n \to \mathbb{R}^n$ defined by
$$
\Phi(y) = M^{-1}(N y + b),
$$
and so $x_{k+1} = \Phi(x_k)$.
$\Phi$ is called a contraction mapping if there exists a constant $0 \leq L < 1$ (called the Lipschitz constant, or contraction factor) such that
$$
\| \Phi(y) - \Phi(z) \| \leq L \| y - z \|.
$$
Intuitively speaking, this means that the mapping $\Phi$ takes any two vectors and moves them closer together. Note that
\begin{align}
\Phi(y) - \Phi(z) &= M^{-1}(Ny + b) - M^{-1}(Nz + b) \\
&= M^{-1}N(y - z), \tag{$\ast$}
\end{align}
and so we have that $L = \|M^{-1} N\|$.
Now, let's relate this to the convergence of the iterative method. Let $x$ be the solution to $Ax = b$. Notice that $x$ is a fixed point of this iteration:
\begin{align}
\Phi(x) &= M^{-1}(Nx + b)\\
&= M^{-1}((M-A)x + b) \\
&= (I - M^{-1}A)x + M^{-1} b \\
&= x.
\end{align}
Combining this fact with equation ($\ast$), we have
\begin{align}
M^{-1}N(x_{k} - x) &= \Phi(x_k) - \Phi(x) \\
&= x_{k+1} - x,
\end{align}
and therefore, defining the error of the method at step $k$ to be $e_k = x_{k}-x$, we have the following recurrence for the errors:
$$
e_{k+1} = M^{-1}N e_k. \tag{$\ast\ast$}
$$
So, we certainly have that
$$
\|e_{k+1}\| \leq \| M^{-1} N \| \| e_k \| = L \| e_k \|,
$$
relating the convergence of the iteration to the contraction factor of the mapping.
In your question, you mention the spectral radius of the operator $M^{-1}N$. The condition $\rho(M^{-1}N)$ indeed is necessary and sufficient for convergence of the iteration, which can be seen by applying the power sequence theorem for the spectral radius to equation ($\ast\ast$).