It is probably quite rare for a lathe headstock and tailstock not to be reasonably co-axial. It would take severe wear, or particularly poor manufacturing standards, to create much discrepancy, assuming the components are original. It is not very important for spindle turning anyway, if you are turning by hand you automatically compensate for any error.
But when turning an item on a faceplate or in a chuck, with tailstock support, alignment is much more critical. If the point of the tail centre is not lined up properly, it wont support the work, it will score circles on it and try to pull it off its fixing.
Many woodturning lathes have a swiveling headstock for larger pieces, and for easier access for faceplate work and hollowing. A very useful feature. But not all are indexed to make sure the centres line up again, or if they are, it is not always very accurate. There is an easy way to remedy this.
Line up the headstock as accurately as you can and lock it down tightly. This may be the last time you have to do it, so get it right!
Drill a horizontal hole for a locating pin right on the joint between the upper and lower swivel plates, parallel to the mating surfaces. The hole will be half in the upper plate and half in the lower. Take it easy in case you break the drill bit. Open up the hole in stages, using a bigger drill bit each time. For maximum accuracy, drill the hole slightly too small, check the alignment hasn't changed, and then use a reamer to size the hole to fit the pin. 1/4 inch (6mm) diameter and 3/4 inch (18mm) deep is about right. Grind a small chamfer on a bit of steel rod, which must be a snug fit in the hole, and fit a handle.
On some lathes, instead of drilling on the joint, you might have to drill a vertical hole right through the base of the headstock and the lower swivel plate. If you use this method, the further the hole is from the axis of rotation, the greater will be the accuracy.
Now you can swivel the headstock without having to worry about lining it up accurately afterwards. Just insert the pin and lock down. Quick, easy, and spot on every time!
Aligning my Tyme Classic headstock