Adjectives that should be adverbs
She drives really good is wrong because good modifies the verb drives and thus should be the adverb well. She drives real well is wrong because real modifies the adverb well and thus should be the adverb really. She drives real good is, of course, a double error. Most of us are unlikely to use adjectives as adverbs except when being deliberately slangy.
Note that I drive slow in town is not an error. Some common adverbs have two forms; both slow and slowly can be adverbs, though the only adjectival form is slow. Don't automatically correct an "adjectival" form that seems idiomatic as an adverb; check the dictionary - it may be a legitimate adverb too. In fact, real is very frequently an adverb in casual speech and is accepted as such by dictionaries - it means very rather than genuinely or veritably and hence is distinct from really - and therefore she drives real well, condemned in the preceding paragraph, has been granted some license.
Adverbs that should be adjectives
I feel badly about it is such a common error that some authorities accept it as idiomatically correct. The verb feel is a linking verb in this example, not an ordinary verb as it can be in other sentences, such as I feel strongly about it and We feel similarly about it. A linking verb links its subject to the following word or phrase. I is a pronoun and cannot be modified by or linked to an adverb, but it can be modified or linked to an adjective. Thus it should be I feel bad about it. An occasional expression such as I feel badly about it may infiltrate the speech and writing of those who are careful of their grammar and know something about grammar but not quite enough; they think the verb feel has to be modified by an adverb, so they tack on the ly. To avoid such errors, pay close attention to sentences that contain linking verbs. The most common linking verb is, of course, be. Other common verbs that can be linking verbs include seem, appear, look, become, grow, taste, smell, sound, remain, and stay.
Precision with adjectives and adverbs can be important. In opening his poem on his father's dying with the line Do not go gentle into that good night, Dylan Thomas was being precise. He wanted his father to remain himself as he faced death, not to be gentle and resigned, but he did not want his father to die ungently and painfully, which is what Do not go gently would mean. To communicate his meaning, Thomas used gentle as what is called a predicate complement - a construction that is quite common, as in I came home tired and Don't go away mad, and is not likely to give any fluent user of the language trouble. However, such usage does surprise us and make us pay attention when we find it in Thomas's line where we would expect an adverb.