Long considered the canonical badge of nerdiness, a pocket protector is a white or clear plastic liner intended to be inserted into a shirt pocket. Ink pens, especially cheaper, ball-point ones, have a reputation for leaking at inopportune times, sometimes resulting in an embarrassing black, blue, or red stain on one's shirt. The pocket protector addresses this issue by providing a barrier for the fabric of the shirt from the ink.
To the average person, this is a silly response to a silly problem. Most people don't feel any real need to carry around a pen wherever they go, and those that do would rather just buy a higher-quality pen than slide a tacky-looking insert into their shirt pocket. By stereotype the people who did tend to carry around cheap pens, and thus pocket protectors, were boffins, engineering students, and other such nerdy characters. The nerdiest of the bunch would carry around several pens of various colors, and maybe a pencil and a permanent marker to boot, resulting in an overstuffed shirt pocket.
Other articles of clothing that tend to accompany the pocket protector are black plastic framed glasses (optionally repaired with tape), a short-sleeved white dress shirt with a tie, pants riding too high on the waist and too short in the leg, and white socks with dark shoes.
Incidentally, a computer specialist at my college was infamous for wearing an overstuffed pocket protector, and one day he leaned over underneath a desk to check a networking cable. When he straightened back up, one of the pens caught on the edge of the desk and catapulted the dozen or so writing instruments all over the room.
A pocket protector looks worst when used with a nice white shirt, the tackiness of the cheap plastic sleeve contrasted against the sort of nice dress shirt mandatory in an office environment is a classic example of the utter lack of fashion sense nerds tend to have. Pocket protectors are, however, generally considered acceptable when used in the shirt pocket of a work uniform or jumpsuit. Mechanics, electricians, and other such maintenance professionals often need to take notes when doing work on complex machinery, and there is no blatant style contrast between a work shirt and the pocket protector in this case.
Pocket protectors are a much rarer sight today than they were twenty or so years ago. The modern equivalent seems to be the Leatherman multi-tool, which also indicates a need on the bearer's part to be perpetually ready to do something nerdy.