People seem to use the term distortion to loosely describe that a lens does not create a representation of what they see in the way that they'd expect. In a colloquial sense this is acceptable, but from an image quality perspective distortion refers to something more specific.
There are a number of factors which can account for a subject not appearing the way one expects in a captured image. I have created a reference list of terms and explanations for each below. It is important to have a common understanding of these terms in order to have any sort of meaningful discussion.
Projection: How the 3D space in the scene is mapped into a 2D image
The lens gathers light and focuses it on the sensor. The geometry implemented in the design can vary. An ideal rectilinear lens design will project straight lines which are in the scene as straight lines on the sensor. An ideal stereographic lens design will project spheres which are in the scene as circles on the sensor.
Keystone: Things which are in projected planes that are closer to the lens look bigger
This is a very simple concept. If you take a wide angle image of a tall building by standing at the bottom and tilting up, then the bottom of the building is much closer to the lens than the top of the building is. Things which are closer to the lens look bigger than things which are further away. So the top of the building will look tapered.
Perspective: The position and orientation of the camera when capturing an image
Perspective is the camera's point of view; its position relative to the objects in the scene as well as which direction it's pointing. If the camera is very close to one object but very far from another then their proportions in the captured image will not represent their relative sizes in real life. If these two objects are the in the same projection plane then their relative sizes will remain in proportion. A camera's perspective can introduce keystone into an image.
Distortion: Differences in magnification of objects which are in the same plane of projection
Now onto distortion; the cuss-word which afflicts so many wide angle lenses. Distortion is a failure to deliver on the intended projection used in the lens design. If a lens is a rectilinear design but fails to project straight lines in the scene as straight lines on the sensor, then there is distortion.
For the sake of discussion I will only refer to the case of a rectilinear lens design aimed perpendicular to a flat surface. There are three categories of distortion, namely: barrel, pincushion and mustache. With Barrel distortion object towards the center of frame are magnified more than than objects towards the corners. With pincushion distortion objects towards the centre of frame are magnified less than objects towards the corners. Lenses with mustache distortion display complex elements of both barrel and pincushion distortion as you move from center of frame towards the corners.
A prime example of a lens which displays mustache distortion is the Samyang 14mm f/2.8. Below is an example from the the cine version of the Samyang 14mm f/2.8 (Samyang 14mm T3.1), which also sells under other brand names in other regions (e.g. Bower, Rokinon).
So in summary, there are some clearly distinctions to be made between the four terms described in this article. Projection is how the lens should render real world objects on the sensor, Perspective is where the camera is positioned, Keystone is a visual effect where things which are closer look bigger, and lastly, distortion is an optical flaw in the lens design.
So next time someone says that a medium telephoto lens "compresses the perspective" please educate them. If you don't have the time or patience then just direct them to this article.