I learned in school that it's correct to say really good. On the internet I've also seen real good. Is this grammatically incorrect, or are there particular circumstances under which this is correct? Perhaps only in American English?
Real is an adjective and really is an adverb. It is not grammatically correct to use these terms interchangeably.
However the adverbial -ly is often dropped in speech, especially for words such as really and badly. For example:
Tom was in a car crash yesterday, he was hurt real bad.
Will often be heard in speech, or seen in informal writing. But to be grammatically correct you would say:
Tom was in a car crash yesterday, he was hurt really badly.
There are also some adverbs which do not sound "right" in speech either and the -ly should be kept. For example:
She carefully walked over the bridge.
As opposed to:
She careful walked over the bridge.
Most adverbs in English end in 'ly', and if you want to be grammatically correct you should use them in full. Using the adjectival form (e.g. quick instead of quickly, rough instead of roughly) is commonly used in informal speech but is not correct.
There are also some irregular adverbs that don't end in 'ly' and may have no relationship with the adjective, such as 'well' in She doesn't drive very well, or 'easy' in Stand easy! Unfortunately there is no rule about these - you just have to learn them.