What is a board-certified vs board-eligible physician?

Passed exam: A board certified physician has passed the examination for their specialty of medicine. A board eligible doctor has completed training in the specialty and is qualified to take the exam.

Passing the exam: A "board-eligible" physician is one who has completed the residency/fellowship training required to sit for the exam. A "board-certified" physican has successfully passed the exam. In most instances, a board-eligible physician has not yet taken the exam, but it may be that he/she has failed it.

Certified or not: Board certified doctors have completed proper training requirements and passed examinations to determine knowledge and competence. If in this process they are termed "eligible". Most who are eligible will in short order get the training and pass the tests to become certified. There is no legal limit to this, so some overuse the term eligible to attain privileges when in fact they never finish.