As stated by Banach fixed point theorem, a contraction mapping has only one fixed point. In plain words it means that the contraction mapping T has only one solution that satisfy $Tx = x$.
A stupid question came to my mind : is it just another way to say that eigenvalue is equal to one for some square matrix T ? If so, why would people build the concept of "contraction mapping" rather than just say so.
Sorry, I am not sure I understood the question, but for a linear map $T$ ,having a unique point is not equivalent to $1$ being an eigenvalue: If $\lambda=1$ , then the associated eigenspace of points {$x: T(x)=x$} is a subspace, and so it will be of dimension one or higher, so it will contain uncountably -many points that will map to themselves, i.e., infinitely -many fixed points, so if you want a unique fixed point then $1$ cannot be an eigenvalue. And, as far as contractions, you need to have the structure of a normed/metric space to talk about contractions. Notice too, as someone wrote in the comments,that contraction maps may not be linear, so they cannot necessarily be represented by a matrix.
Maybe the bottom of this link http://mathworld.wolfram.com/LinearTransformation.html
Will help answer your question.