According to the Merriam-Webster dictionary, "atypical" is someone or something unusual or irregular. Doctors use the term to describe diseases that cannot be classified in existing categories, such as "the patient suffers from atypical depression." It is wrong to use "informal" as a synonym in atypical use of language; informal language is a change in the way you talk and the vocabulary you speak -- not systematic wrong use of the language.
People who learn a language other than their mother tongue can confuse the rules of grammar and the meaning of different words, creating sentences that make little sense to fluent speakers. For example, double negation in English, such as "I don't know nothing," is wrong, but in Greek, it is wrong not to use double negation. Therefore, an English speaker using English grammar to create Greek sentences is bound to form meaningless phrases.
Young children who haven't gone through the education process yet and people with abnormal language development may use the language in an atypical manner without knowing they are doing so. Abnormal language development can be a result of disorders that have an effect on people's capacity for learning, such as autism and hearing impairment. However, it can also be a result of a child having to follow her parents and attend schools using different languages -- as is the case with international migrants -- or being deprived of formal education completely.
Atypical language can be used by speakers who are aware of the language's proper rules. In this case, atypical use is deliberate and happens because all interlocutors consent. For example, when British teenagers talk to each other, they can use "innit" instead of "isn't it" at the end of a sentence, even though they are aware that the latter is correct. Text speak, like "c u l8r" instead of "see you later" is also an example of deliberate atypical use of language, as writers aim to send the SMS quickly or fit its contents into a single message.