I keep seeing references to Vertical and Horizontal Drop Outs, and I really do not understand what they are,

Anybody have pictures of both so I can understand this?

Sheldon to the rescue: https://sheldonbrown.com/gloss_dr-z.html#dropout

(you may need to scroll down for pics)

Bob;

Thanks, that's what I was trying to understand,, now it all makes sense.

Why did Sheldon say the track "fork end" is not a dropout? Just because the wheel can't drop out?

He's being pedantic. Everybody else calls them "dropouts." When you go to buy them from frame supply shops, you find them classified under "dropouts." Just call them dropouts and everybody will know what you mean.

In regards to the horizontal dropouts; what is the "correct" position for the rear wheel? As long as I have been around bikes, I have always moved the wheel as far back in the dropout as I could. But, others have made mention of "your wheel is too far back"; and the adjustment screw found in some implies that it should not go all the way back.

So where should it go? As an experiment, I moved my rear wheel to where it was at the intersection of the seat and chain stays, and it is easier to shift onto the largest rear cog when the chain is on the big chainring (it is an early 70s ten speed.)

-James

If you have a single-speed or an internally geared hub (i.e. nothing else taking up chain slack) the correct position is when your chain is at the right tension.

If you have a rear deraileur, the correct position is the same position as when you adjusted the deraileur; which is why the world switched to vertical dropouts.

Where you moved your wheel to probably just closed up the deraileur adjustment to closer-to-optimal; you could probably get the same result with the wheel all the way back by twiddling adjustment screws, which I might do since "all the way back" is at least a repeatable position. The adjustment screw some have lets them pick a different repeatable position; exactly where the wheel goes has some effect on handling, but this is probably more subtle than you care about if you ride that bike

Most derailleur manufacturers also offered dropouts with geometry engineered to optimize the performance of their derailleur. In general, this is achieved when the axle is lined up with the axis of the seat stay.

Vertical dropouts become popular with the advent of indexed shifting. Indexing performance is greatly affected by chain gap which is distance, along the chain, between the points where it contacts the cog and jockey pulley. By eliminating the fore/aft axle positioning which affects chain gap, engineers were able to eliminate one variable that the home mechanic could inadvertently disturb and upset the performance.