The large mandrel has a small spiggot which measures 18mm and the larger size which is 25mm and is for use in your chuck.
The smaller mandrel is for use in a drill and can be used to buff an item while it is still mounted on the lathe and it has a spiggot of 10mm and a shaft size of 13mm. It can also be used to good effect in a pillar drill.
Before you can use it you obviously have to mount it on you lathe, I have Administer chucks and a variety of jaws but none of then has either an 18mm or 25mm diameter grip. They were all one or two mm larger or smaller but I found this was not a problem as they still gripped the mandrel properly and held it firm during use.
The important thing when installing the mandrel is to make sure it runs true. The best way to do this is first make sure your head stock and tail stock are aligned properly. Then put the mandrel in the chuck and grip it lightly then bring up the tail stock with a center installed into the end of the mandrel and tighten the chuck, as below.