A sore is a broken patch of skin — something you might put a bandage on — or it's something that gives you pain, like a sore throat, that’s red and scratchy and painful.
You can have a sore and you can also be sore: if you've just ridden your bike over the Alps, you're probably pretty sore — achy and hurting all over. Maybe you've heard phrases like "Don't be sore at me, boss," from old gangster movies, meaning "don't be angry." A sore point or a sore subject is something you don't want to discuss, like the sensitive topic of your last history quiz, the one you slept through.