Asynchronous is a form of communication that is turn-based, not real-time. Letters and emails are asynchronous while phone calls and instant messenger chats are synchronous.
Originally, asynchronous learning referred to a style of formal learning where the student and instructor has regular interactions, but the interactions were separated by delays of hours, days, weeks, or months. Most early post-office based distance correspondence courses followed this model.
This "instructor supported" asynchronous learning is still used today, although technology has reduced the "turn-around" time for an instructor response more towards minutes, hours and days, rather than weeks or months.
Increasingly, asynchronous also means a type of learning that is self-paced. The subject matter expert, instructional designer, and technical designer create the content, such as a paper or web based workbooks or videos; the end-learner consumes the content, even completes quizzes, without tailored feedback from another human.