When I’ve got a jar that just won’t open, I tap the outside rim of the lid with the handle-end of a butter knife. Most of the time, this will break the seal and I’ll be able to open the jar easily.
If it’s jelly or some other sticky kind of thing, I’ll also run the top of jar under hot water for a minute. Sometimes some of the sticky stuff got caught in the threads, and the heat melts it, enabling it to open!
Another thing to try, if you’re having trouble getting a grip on it, is to wear a rubber glove. It works like one of those rubberized jar openers, except it does double duty when you’re cleaning.