An elucidation is really just a fancy name for an "explanation." Sherlock Holmes revealing his deductions to Dr. Watson is a classic example of elucidation.
Because of its suggestion of complexity, elucidation is a word often used in a scientific or educational context: scientists will elucidate their latest theories to the public or a college professor will elucidate some complicated theorem to their students. You need to be wide awake for an elucidation. There's often a test to follow.