As Sith Lords, Darth Vader and his master Emperor Palpatine clashed several times in the established Star Wars canon (with varying results).
The Skywalker Saga films don't make it all that clear how Palpatine survived after Return of the Jedi. For that, we'll have ...
The apprentice who kills Plagueis that story is referring to is meant to be Palpatine, Darth Sidious. But before Sidious, the idea appears to be that during his artificially extended life ...