Could be a loose bearing mount, worn bearing, damage to the spindle itself, loose screws in the headstock mount, loose motor mount, loose pulley mount, damage to the belt, damage to the handwheel, damaged motor, ect ... many things could be wrong with it.
Pictures would probably help a lot.
But first things first, check to see if the headstock is secured to the ways properly, then check the motor mounts and the belt and pulleys ... hopefully it's something quick and simple in that direction. Loose pulleys are simple ... just tighten the set screws once you've put them back in their proper positions. A wobbly handwheel could also be the culprit, make sure it's set screw is properly tightened too!
If it needs to be secured back down, you'll want to create or obtain a taper alignment tool so that you can line the headstock back up with the tailstock as you tighten it back down. If it's the pulley belt, obtaining a replacement shouldn't be all that difficult, and putting it on should be fairly straightforward.
If it's not one of those issues, then you may be dealing with a damaged spindle or bearing, which are a bit more involved for repairs ...
You'ld have to pull apart the lathe headstock to get at the spindle bearing mounts to examine them ... if one is loose, you can tighten it back down to see if that fixes the problem. If not, you may be in for a bearing replacement or a shaft replacement. Make sure that the shaft is running straight and true and there's no damage to the shaft itself that could cause vibration against the bearings. If none of those are the issues and you are getting wobble, your bearings or bearing mounts are bad. Bad bearings and bad shaft are some of the highest cost repairs among these issues ... test everything else first.
If the motor shaft somehow got bent or damaged, it could also cause your vibration issues ... replacing the motor itself is the single most expensive issue you'ld have to fix, but checking it is relatively easy. Just look at the shaft with the pulley on it and turn the lathe on ... it should run smooth, with no wobble or vibration in the pulley.
