English teachers used to drill into students that they did not "feel good." They "felt well." It was the corollary to "I feel bad," not "I feel badly," to which many teachers would reply something like: "Well, maybe if you took off your gloves, you could feel better." "Good," "well," "bad," and "badly" all define how you feel, but not in the same way, grammatically.
But things are changing, for good and for bad. "I feel badly," using the adverbial "badly" instead of the adjectival "bad," is heard (or written) frequently enough to have reached Stage 2 on the five-stage Language-Change Index in Garner's Modern American Usage. That means it's still wrong, grammatically speaking, but less wrong than it used to be. You probably won't flunk if you use it.
Interestingly, while it used to be considered improper to use "badly" when you meant "a lot," as in "badly in need" of something, that usage is now considered standard English. And dropping the "ly" from "bad" when you need something a great deal is also at Stage 2, so it's more proper to say "I need chocolate real bad" than it used to be. But because so many people will turn noses up at hearing that, if you really want the chocolate you might be better off saying "I badly need some chocolate." If you are of the belief that adverbs can be split from their verbs, you might say "I need some chocolate badly."
Now, to the opposite side. It's another quirk of English that the "ly" forms of "bad" and "good" are not the same: "Badly" is an adverb means "worse" or "incorrectly," while "goodly" is an adjective meaning "in abundance." Though people frequently use "badly" in ways that teachers would disdain, almost no one uses "goodly" incorrectly.
It used to be, though, that if you said "the band in the parade played good," you would have had your knuckles rapped by the same teacher who told you that you feel "bad." The word you wanted, she would admonish, was "well," the adverb, not "good," the adjective. But Garner's says that using "good" instead of "well" is gaining acceptance. If you're talking about personal performance, "good" is at Stage 2. But if you're talking about the performance of inanimate things, surprisingly, it's more acceptable to use "good," as in "the engine runs good." That usage is at Stage 3, meaning you might get only a black mark in the teacher's book, not a trip to the principal's office.
It's all well and good to know when you're being bad.