Polygon

A polygon is the basis of all 3D rendering. It is defined by the following:

Plane: any flat level that resides in 3D space
Vertex: the point in 3D space. A corner of a box can be considered a vertex.
Edge: the line segment between exactly two vertices. A piece of paper has four edges
Polygon: an enclosed 2-dimensional shape with straight edges. Usually a triangle, but can have more.
Face: more or less a synonym of polygon. The difference usually comes about when one speaks of a brush face being split into several polygons by the build process.

Unreal Polygons

Note that these might not be the proper technical names, they're just what I call them because I haven't seen anywhere else what they should be called – tarquin

Wireframe Polygons

Also known as a brush
Always convex
Can be merged with intersecting

Surfaces (Brush polygons)

The face that is selected when clicking in the 3D UnrealEd Viewport.
Each surface can have its own attributes controlled by the Surface properties window.
The number of these is given by the polycount.

These can differ from wireframe polygons when a brush has had the Merge Polygons command applied. These are not necessarily be convex. In fact, a single selectable surface can be formed from several entirely seperate pieces.

Nodes

Created by BSP build, sometimes called BSP polygons. Always convex. Often a subdivision of Brush polygons. These are counted by node count.

Comments

Nytro: Okay, I've been quite confused on this subject for a while. In modeling, a polygon refers to a 3-sided surface that makes up part of the model. In UnrealEd, it's sometimes refered to as some convex thingy? And sometimes just a surface?

Mychaeel: As a term, "polygon" implies that it can have more than three sides. For that matter, I had thought even in modeling the correct term for what you're describing was "face" (or perhaps even "triangle"). – In UnrealEd, a polygon must be planar and convex, but it can have any number of sides.

Foxpaw: Mmm... 3D modeling programs these days seem to mostly support only 3 sided polygons unfortunately. Rhino supports any number of sides, but this can cause havok if you try to export to another program. The confusion is certainately understandable.

RDGDanClark: As far as I know, any 3D modelling program will support polygons with any number of sides, but this is just not optimal for performance. Rendering applications like the 3D objects to be triangulated, i.e. divided up into triangles.

Foxpaw: Well, the programs can usually make polygons with any number of sides but I don't think they can usually export to a common format like that. In my experience anyway, exporting to DXF, LWO, or 3DS always seems to trianglify the mesh "for me" against my will.

Sobiwan: This conversation is good and might be better suited in a general modelling page. The closest I found was the Unreal Modeling FAQ.