Formal Language refers to speech or writing that uses standard, established words, and follows the rules of grammar. Sentences that are written in formal language are typically complex and they do not include contractions like "don't" or "they're." Often students think that formal language is "old" language, but actually, this is not the case; formal language is not just the language of Shakespeare (who in fact uses a lot of informal language in his writing!) You should always use formal language in your academic writing, except for creative writing, but you should check with your instructor first to make sure that it is appropriate.
Informal Language refers to slang and colloquial speech or writing. Informal language can be short and choppy, and does not necessarily use a variety of sentence structures -- in other words, informal language is often repetitive. Informal language is the type of language you are likely to use when talking to friends or communicating on social media. You should not use informal language in your academic writing, but when you read fiction (novels, short stories, etc.) you are likely to see informal language.
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Image Credit: Language by Corpus Deliciti from the Noun Project