In respectful disagreement with post#2, academic English is intended to be more precise and clearer than conversational English, not stilted. Clear pronoun reference is demanded, idiomatic expressions or colloquialisms or slang or contractions are not allowed--all of which can be ambiguous to other readers who are not from the same region as the writer. In short, academic writing demands clarity and coherence so that effective communication is effected. Rather than using "big words" to merely be pedantic, academic language demands precise words, words that truly denote the meaning of the writer.
Academic writing is expository writing, and, therefore objective. Thus, it demands the use of the third person rather than first or second. Because it is used to analyze and explain, narrate or persuade, semiformal language is necessary.
Formal language is the highest level and is rarely used anymore in the U.S. Examples of this level of language are the U. S. Declaration of Independence and other documents, such as the Gettysburg address.
Academic English has to be written in a very stilted style. It cannot sound anything like what a person might say in conversation. It needs to include sentences with many clauses and (as the previous answer says) lots of big words. The more academic the audience, the more this must be so. Academic English is not particularly aimed at being easy to understand. Instead, it is meant to sound extremely formal.
Well, you might find that there is some debate abotu this, as every teacher might have different guidelines on what they mean when they say they want an assignment completed in "academic English." I think, however, for me, my list of what is meant by academic English would include the following:
1) English that is written without the use of slang phrases and contractions with a higher than normal level of vocabulary. This means there are no repeated phrases that we use in speech like "You know" and "like." It should not read as being conversational.
2) English that is written in the third person (not "I").
3) English that is well structured into an essay or report with a clear introduction and conclusion.
4) Writing that includes references as appropriate.
I think probably that is my list, but others might want to contribute. The biggest thing that I find my students struggle with is their level of vocabulary and writing in the third person, which you need to practise on to master.