Can someone explain poly counts to me at a simple level?
'Polygons' refers to the triangles that every 3D mesh is made from. 'Poly count' is just the total number of triangles. In professional circles, they call them tris, but everyone around Trainz still calls them polygons. Polygon of course just means 'many sides', whereas a tri is more specifically a 3-sided polygon.
How does it affect images?
Poly count doesn't affect images, if by images you mean the textures on the 3D mesh. It can affect the appearance/realism of the object - too few polys and the object will look blocky.
How does it affect computer performance and the display? Are there rules of thumb on how much a route handle or does it depend on the computer hardware?
For every computer there will be a maximum number of polys in a scene that it can draw and keep track of without the frame rates slowing down. I don't know what that number is, but it will vary with the quality of the video card and maybe with the speed of the hard drive read/writes etc. So, the less polys the better for fast, smooth rendering of scenes. Performance can suffer if too many separate texture images are used on a model - one texture is worth about 200 polys in terms of the computer 'resources' needed to handle it. Good modellers strive to stick all their textures into just one image, but sometimes you have to use 2 or 3 due to the technical requirements of the texture mapping. More textures than that is a sign of a slack/ignorant modeller. I think the record is 50 textures on one model - really bad.
How can you tell how many poly counts or whatever you call them each item has? How can you tell which items have higher poly counts than others?
There's no way to tell just by looking at it. However, in Content Mangler, select an asset and I think a right-click will show a menu which has an item called something like 'mesh technical properties'. This will list various parameters including poly count. Or, if you can open the asset's folder and see the mesh file itself (it will be either a .im or .pm file) you can use PEV's "Mesh Viewer" utility (a freeware program) to open the mesh, right-click and view 'mesh data' to see the poly count. I think both methods can also tell you how many different textures are used on the mesh.