Looking for a Description of how the Map is Shown on a PC (rendering process but no technical details)

I'll have a go...

At it's heart, OpenStreetMap is just a big pile of data that represent things in the physical world such as roads, trees and all sorts of other things.

The way that the data is organised is described here and data items have tags with values that say what the data represents. Lots of information about the sort of data that is stored is available on the OSM wiki and information about what tags people have used is also available on pages such as this one.

When you point a web browser at http://www.openstreetmap.org/ you're seeing a representation of that data (one of the the five that are available there) - click the "layers" button at the right to see the others. Behind the scenes the data is converted to an actual map by a process called rendering. The actual maps that you see on OpenStreetMap.org are made up of individual tiles such as this one. Most of these are actually created only when your web browser asks for them, from the latest OpenStreetMap data.

If you want to set up your own map server that you can point a web browser at you can, if you follow the instructions here. You can use the same style as the "standard" map on osm.org, or the humanitarian style or you can use these or others to create your own web map style.

Lots of people have created lots of maps in different styles showing different sorts of data.

A web map is only one one sort of place where you'll see OpenStreetMap data. You can download maps for handheld GPS devices and car sat-navs, (or create your own), for mobile phones and some services using OpenStreetMap data don't involve maps at all.