Start with a pseudo-Riemannian manifold , let be another pseudo-Riemannian metric on , we say that and are conformal if there exists a positive scalar function on such that (sufficient smoothness of the relevant quantities are always assumed).
Observe that two conformal metrics measure angles the same way: recall that on a pseudo-Riemannian manifold , given a point and two non-null vectors , the angle between the vectors can be defined by
(Notice that on Euclidean space, if form an angle , then .) Thus if is conformal to , they define the same angles
In fact, this inference works the other way too. If , are two pseudo-Riemannian metrics such that for any two vectors we have , then , are conformal (up to a change of sign) by the above definition (see e.g. Exercise 14, Chapter 2 from B.O’Neill, Semi-Riemannian Geometry).
So, in plain English, two metrics are conformal if they measure angles the same way.
Now, let be a pseudo-Riemannian manifold that is non-compact. A conformal compactification of is a choice of a metric such that can be isometrically embedded into a compact domain of a pseudo-Riemannian manifold (well, I am ignoring some regularity issues here). Let be the conformal factor as before. Then observe that any regular extension of to the conformal boundary must vanish on said boundary. This reflects the property of a conformal compactification that “brings infinity to a finite distance”.
The simplest example of conformal compactification is the one-point compactification of Euclidean space via the stereographic projection. In this case, the target manifold is compact itself, taken to be standard sphere. The source manifold is Euclidean space with the standard metric, and the image set is taken to be the sphere minus the north pole.