how does a cigarette lighter adapter work?

Sounds like the fuse holder opened and the fuse and retaining spring fell out. The very end is the contact that goes into the lighter plug and touches the + contact it has to then contact the fuse inside your adapter and then a wire to the item you are powering, or a circuit in the adapter and then the wire to your item. If there is no way to fasten the little silver tip back to the adapter there may have been a nut that slides over it and then fastens to the case of the adapter. The silver tip may be the item you call the metal cylinder. If so you may be missing the fuse. If so you have to buy another. The item you are powering should have a current rating along with the voltage and the fuse must be greater than that rating. Not to much though. As I can't see your adapter and I can't draw pictures on here that is the best I can do for you...

You have an adapter that plugs into the lighter socket. the center probe is positive and the side contact is minus. this comes directly off the positive and negative of the battery. this gives you 12 volts if you have a 12 volt system.