These chucks are held on to the spindle by short bolts or nuts and a captive stud which is fixed into the back of the chuck. Having removed those nuts / bolts the chuck should just pull off in the direction of the tail stock. Little machine shop has a good picture of the spindle here Spindle, Mini Lathe 7x14 - LittleMachineShop.com There are two hole patterns - four at 90 degrees used on the four jaw chucks and three at 120 degrees for the three jaw. The holes are clearance on the studs / bolts. The chuck is registered on the projecting smaller diameter part of the flange integral to the spindle. This projecting flange meshes with a similar recess that is found in the back of the chuck body when you have removed it.
Is the lathe new? If it is contact the supplier as it has either been abused or has rust or dirt trapped between the mating diameters. IMHO this indicates that the machine should be replaced. Do not accept a replacement spindle and chuck unless you are confident you can dismantle and reassemble the beast. Not difficult and here is a link to some instructions from Arceuro. Although the instructions apply to the Seig C3 the method is similar although the castings are slightly different. Arc Euro Trade - C3 Mini Lathe Bearing Change
Hope this helps. By the way if you leave the electronic trip set as supplied it will stop the motor before the plastic gears fail - don't ask me why I am sure but I bought a set along with the taper roller bearings just in case! However the lathe performs well so I am leaving it alone. Guess having the bits in the draw under the lathe is like buying insurance or carrying an rain coat in summer.
Can I ask why you are looking to reset the tail stock as there are several factors to take into account if accuracy is you bag. The centre of the tail stock has to be true for all positions along the bed. This is checked by turning between centres and there is a good write up on how to do this on the web. BUT and it is a very big but some of the correction may involve the head stock as the bed reference 'vee' + headstock spindle + tail stock spindle alignment all have an influence on the accuracy of your machine. If you have just moved the tailstock on the adjusting screws getting it back to the start position is easy. Just do as Dano has suggested or simply put a bit of ground round stock in the drill chuck mounted in the tail stock bring the protruding ground rod up to the headstock chuck with its jaws wide open. Then simply close the jaws a bit at a time testing with a slip of thin paper that the grip is equal. Most likely it will require adjusting but as you get near to the correct setting just rock the chuck back an forward using the chuck key making sure the lathe cannot be powered. Sounds complicated but is very quick to do and is a good use for a broken drill. Also do not forget the jaws on the chuck are designed to be tightened in a set order. Get into the habit of nipping the work with jaw number one and then tightening the others in numeric order. Some chucks are just marked with a punched dot and not numbers - the jaws are also similarly numbered.
Regards - Pat