Basically the hub has a circular pattern of ball bearings inside. On old bikes the axle then has two cones (one on each side) like this
The bearings run around the polished bit at the top.
The play can be caused by not tightening these cones up enough,wear to the bearings or the polished surface itself or wear to the part of the hub that holds the bearings.
I'd try to tighten the cones up to see if you can get no play but with the wheel moving freely.
ps - should of added that on the axle you will normally have the cone and then outside that a locknut - you need to undo the locknut,tighten the cone and then re-tighten the locknut (just on the drive -side).