After reading the previous posts, I at least, have some terminology, and a little more information, however.. . My daughter's boyfriend recently installed a new sink faucet for me, and now the drain tail pipe (PVC) that slides into the compression nut above the s-trap has been separated, several attempts to unscrew and rescrew the nut doesn't seem to work, unless I'm expecting too much from the connection. (Daughter dumped the boyfriend), my question: does the washer fit around the outside of the tail pipe, or does it sit on the bottom of the tail pipe? Then do I raise up the s-trap, slightly and then rescrew the compression nut, or would that cause a problem somewhere else? Should I buy a new compression nut and washer, and does the compression nut slide on, or does it get glued on with the PVC glue? Thanks for your help!

A bit tough to follow.

Put the nut on the tailpiece in the proper direction. The washer should have a connical shape. The cone should face downward.
SO, at this point, the washer will hold up the nut.

Insert the tailpiece into the P trap. Now tighten the nut while pushing up on the trap.

The nut will have to be tightened pretty tight so it doesn't slip.

If the tailpiece doesn't fit far enough into the p-trap, you'll end up with problems too.

All 1 1/2" traps come boxed with two gaskets. One a 1 1/2 X 1 1/2" gasket and one 1 1/2 X 1 1/4" gasket. The BF simply used the wrong one. Go vback and swap the 1 1/2" gasket for the correct one. Good luck, Tom

On top of Tom's and KeepiSimpleStupid's, I am attaching the picture below will give you an idea what they are talking about, may be yours will not be like this Thanks.

John