Conceptually, 3d projection is pretty simple. Mathematically, it's a mess unless you cheat*.

Think of projection like a shadow (or a projector/camera ). If you shine a light on something, it will leave a shadow on a flat, 2d surface that is what you would expect to see on a computer screen. So, in order to view the outline of something, all you have to do is draw straight lines from a point source (used for simplicity) through the object onto the screen, kind of like

Wherever a line (known as a ray) passes "through" the screen, a pixel is drawn.

This method will only draw outlines though. If you want to draw the object itself, then another method known as raycasting works. Instead of a light source, you draw the rays from the user's eyes through a middle screen to points on the object. The intersection of the lines with the screen is where the pixels are turned on.

*By cheating, I mean doing anything other than modeling the scene in 3d. Many useful algorithms work around this.

I dont have an example of cheating, but if you go back a ways I have a TI-Basic implementation of 3D wireframe projection.  It's slow, but accurate.  The Wikipedia article on 3D projection is very useful.

One way of cheating is using right triangles, but with 3D instead of 2D.  That way, you can use proportions.

Thats not cheating, thats using accurate math to get real projections of 3D points onto a 2D plane

Things like isometric displays, 1/2 point projection, raycasting, or anything else that does not give true 100% control 3D.  Be is as it may you can get quite realistic images through cheating (video games today still cheat on many things).  Basically as i see it, if you can view an object in 3D space and have complete control over rotation, translation, and scaling, you are working in true 3D.  Whenever you add constraints for the purpose of increasing speed, i see that as 'cheating' a little bit since its not true 100% 3D anymore.  Note that i don't mean cheating in a negative connotation Like i said, there is not a game out there on the market that does not cheat.  The only way (Imho) to not cheat in a full image render (with objects, reflections, light, texture, ect...) is to raytrace from the light sources, calculating the reflections and refractions, and calculating exactly how the human eye would perceive the scene.

Ah yes you are correct, its much more efficient that way

Take a look at prgmRENDER in my TI-Basic 3D demo.

Here are the formulas i use:

Sx = S*(X/Z)
Sy = S*(Y/Z)

Where (X,Y,Z) is the coordinate of the point in 3D, S is the "size" of the screen, and (Sx,Sy) is the coordinates of the point on the screen.  To account for rotation, you would rotate the coordinates (X,Y,Z) before displaying them.