Overhauling my 83' Raleigh I found a spacer in the rear dropout on just one side. This prevented the wheel from sliding all the way back into the dropout. I've taken the spacer out and the bike seems to work fine and also its easier to center the wheel within the seat stays because I just pull it back all the way into both dropouts. Using just the one spacer I need to pull back on that one side then carefully try to match the other side while tightening the skewer. Can anyone tell me why I need this spacer? Should there be one on each side? And which side should it go on the drive side or opposite? THanks-Joe

You don't need the spacer. It's just there as a convenience for those who want the wheel farther forward, and usually used in pairs. Also on many bikes the wheel doesn't center correctly if the wheel is all the way back on the dropouts, you're just a bit lucky that way.

If it works for you this way, you're set. If you decide you'd like the wheel farther forward, and have the convenience of automatic centering, buy another spacer and set both for centering in the right position.

I have seen some bikes that are configured to be one link too short in the chain if the wheels are back in the drop outs all the way. Some bikes even come new out of the box like this. So, beware. Check to ensure the chain is long enough in big/big while the wheel is full seated back in the dropouts before riding!!

Another aspect of where in the drop out slot the wheel should be placed is the shift performance. Often, somewhat dependent on the chain length, is that the further back in the slot the axle is clamped the greater the distance from the der's upper pulley and the cogs underside. More cog/pulley gap usually means more sluggish shifting. Andy.

Some rear derailleur hangers take some space in the rear of the right dropout. Perhaps your spacer was inserted when such a derailleur was present to take up a similar amount of space so that the wheel would sit straight when pulled all the way back on both sides.

My thoughts initially, but all bolt-on hangers work the same way, so if he removed it, how/where would the OP attach the RD. Based on that I suspect that he had a "simplex" or similar wheel stop in a conventional dropout. These were used in pairs to serve the same purpose as the micro-adjust screws that Campagnolo popularized.

I think the axle should be positioned where the tubes of the rear triangle intersect, and the spacers should be there to accomplish this.

I think the axle should be positioned vertically in line with the derailleur mounting bolt.
While the spacer is not required, having the wheel all the way back in a horizontal dropout is not usually the optimum position. Many bikes came with one or two of these spacers , especially on stamped steel dropouts that don't have adjusting screws. The OP did not mention which side it came off of but Ihave seen them installed on either or both sides.

Well bolt on derailleur hanger and a piece as well added to the left,
in combination, helps you center the wheel again,
in puncture repairs is convenient,

and If you like the index shifting, one of the things that make them work
is getting the rear axle in the same spot.. consistently..
hence predominant vertical dropouts..

Those spacers are often found on bikes using s "claw" type derailleur mount, to facilitate centering the wheel between the stays. They aren't strictly necessary; only a convenience.

Gotta agree with you - and I've seen them mostly on the LHS. (completely different from the spacers that are an integral part of claw type derailleur hangers which of course go on the RHS). These show up more on bikes that have their own integrated hanger.