How can I repair or remove my bathroom sink's tailpiece?
The typical setup here is you have


The top collar
A rubber gasket or sealant (i.e. plumber's putty or silicone)
The sink holes
A rubber gasket
A compression collar


The rubber gasket can be a fiddly thing, where you need to seat and re-seat it to get a positive seal. If you're having trouble with it (and based on the pictures I'd say that's likely) try coating it in petroleum jelly. It should help if you have any odd spots in the sink hole that the rubber can't fill