A distance matrix is a matrix containing the pairwise distances between the elements of a set. The "distance" may rely on the concept of metric.
The distance matrix for an N-elements set is a NxN matrix in which entry (i,j) contains the distance between items i and j.
The distance might be related to the concept of metric (e.g. Euclidean distance, Mahlanobis distance, and so on) but there are also non-metric distance matrices: an adjacency matrix is an example of such non-metric distance matrices.