I have a lot of data consisting of points in $\mathbb{R}^{100}$. I would like to represent these points in two or three dimensions, in such a way that roughly preserves the distances: i.e., if a group of points forms a cluster in $\mathbb{R}^{100}$, they will form a cluster in the visual representation; if they are far apart, they will be far apart in the visual representation. Of course, the picture cannot be perfect, but I want something that provides a nice illustration for the casual observer. What are some common ways to do this?
Edit: I am not fussy; even an algorithm that inputs many points and their pairwise distances and outputs a visual representation would suffice.
Also: here is an example of the type of picture I wish to generate.