This is commonly referred to as the "eye-bag surgery". With age, the fat in the lower eyelid starts to bulge. Removing it give a younger appearance. Not every lump in the lower eyelid is due to fat. Some people have a strip of prominent muscle of the lower eyelid, called the orbicularis oculi. This strip of muscle cannot be removed.
There is also something called a "tear-trough deformity". The tissues of the mid-face descends with age. The bony socket of the eye is exposed, creating this. appearance. Removing the bulging fat from the lower eyelid does not make the "tear-trough deformity" disappear. The bony socket of the eye needs to be covered / camouflaged.
A curved lower lid makes a person look old. This is sometimes called "senile ectropion". Tightening the lower lid during eyebag surgery gives a better result.
Compare the above photo with a young person's eyes.