Plain English Meaning
These are written, signed notes the instructor puts in the student's logbook to confirm the student has been trained for and is cleared to do something specific — like fly solo for the first time or take a checkride.
Full Definition
Written, signed, and dated statements made by a certificated flight instructor (CFI) in a student's logbook or on a knowledge or practical test application, certifying that the student has received specific training, demonstrated required proficiency, or is authorized to perform a particular activity such as solo flight, a knowledge test, or a practical test. Endorsements are required by Federal Aviation Regulations and serve as the official record that the instructor has met their regulatory responsibility for that authorization.