This procedure usually consists of removing herniated disc material that presses the nerves or spinal cord. Discectomy is a decompressive type of surgery.
Hyperextension means a movement where extension is performed at any joint beyond its normal range of motion.
A back extension is an exercise that works the lower back as well as the mid and upper back, specifically the Erector spinae. Each of us have two Erector spinae muscles, one of either side of the spine that run along the entire length of the spine. Erector spinae muscles are actually formed of 3 smaller muscles - spinalis, longissimus, and iliocostalis.
The name hyperextensions are being used for back extension exercises that are done using a hyperextension bench in a fitness gym. However the name 'hyperextensions' is a misnomer, because what you are trying to do here is only to extend the spine within its normal range and not beyond its normal range of motion. When you extend the back from the flexed position, at the end range, your head and neck stays in neutral position.
In fact, back extension beyond the normal range of motion has been found to be detrimental for the exerciser. Hyperextensions during dead lift have been found to lead to lumbar disc pathologies and muscular spasms.