If you are 'handy' then it's a simple procedure to adjust, but I'll let you know it may be new wheel time.....
Without knowing the 'quality level' of the wheel, the bearings should be smooth to turn and no play.....
Do you know the make of the 'hub' and any serial numbers..e.g. like Shimano M525...
If you can get away with re-greasing and adjusting the hub 'cones' and bearings then OK (you'll need special thin spanners) then OK - but I'd guess on the way out unless you know about bike bits........ (i.e. budget a new wheel).....
If you are not sure what's going on, then maybe a bike shop..... ? Wheel cost goes from...XX to XXXXXXXX......?
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).
I'd do it on the non-drive side, then you don't need to adjust your gears.
Undo the locknut, tighten the cone by screwing it in along the threaded axle a little, try to rotate the axle and if it won't rotate or feels rough then you've over-tightened so back it off a bit, then when it's correct tighten the locknut against the cone by screwing the locknut in and the cone out so they tighten against each other.
You'll almost certainly need cone spanners, which are thin spanners to fit the flats on the cones, where conventional open-ended spanners are too thick.
They come in pairs generally, fit different sizes, and bike shops sell them quite cheaply.