What is the difference between Part 141 and Part 61 of the FAA Regulations?

When training under Part 61 regulations any FAA-approved flight instructor, whether associated with a flight school or not, may train a student.  When training under Part 141 regulations only instructors associated with an FAA-approved flight school is permitted.  Both methods of flight training require the student to meet the same standard of performance in order to obtain a pilot certificate.