Although, PCA tries to use optimization for retained variance, multidimensional scaling (MDS) tries to retain the relative distances as much as possible when reducing the dimensions. This is useful when we have a high-dimensional dataset and want to get a visual impression.
MDS does not care about the data points themselves; instead, it's interested in the dissimilarities between pairs of data points and interprets these as distances. The first thing the MDS algorithm is doing is, therefore, taking all the 
datapoints of dimension 
and calculates a distance matrix using a distance function 
, which measures the (most of the time, Euclidean) distance in the original feature space:
Now, MDS tries to position the individual datapoints in the lower dimensional space such that the new distance there resembles the distances in the original space as much as possible. As MDS is often used for visualization, the choice of the lower dimension is most of the time two or three...