There is no point that is equidistant from 4 or more points in general position in the plane, or $n+2$ points in $n$ dimensions.
Criteria for representing a collection of points by one point are considered in statistics, machine learning, and computer science. The centroid is the optimal choice in the least-squares sense, but there are many other possibilities.
Added to answer the comment:
The centroid is the point $C$ in the the plane for which the sum of squared distances $\sum |CP_i|^2$ is minimum. One could also optimize a different measure of centrality, or insist that the representative be one of the points (such as a graph-theoretic center of a weighted spanning tree), or assign weights to the points in some fashion and take the centroid of those.