Obviously, the underlying principle here is that the mandrel must match the spindle on your lathe. You can choose either a threaded mandrel (3/4"x16tpi or 1" x 8tpi are the most common choices), or you can choose a morse taper mandrel (MT1 or MT2).
But a potential concern is that for some new turners, the specifications of the lathe are unknown. Now, that might surprise a few folks, but I know that the specification for my lathe that appears in both the retailers advertisements and the instruction book that came with the lathe, is wrong. I ordered a threaded stopper mandrel from PSI based on the specification. But when I went down to the shop a few hours later and looked at the lathe, I recognized that the specification was wrong. Fortunately, an e-mail to PSI customer service resolved the immediate problem. I later sent an e-mail to the lathe manufacturer who acknowledged the error, but a year later the same erroneous specification appears in web advertisements for the lathe and also in the version of the instruction book that can be downloaded from the manufacturer's web site.
So the old 'measure twice' rule applies here - before ordering the mandrel, take a good look at the lathe and measure the spindle dimensions to make sure that you are ordering the right tool.