Gig line is a military term that refers to the alignment of the shirt

Shirt

A shirt is a cloth garment for the upper body. Originally an undergarment worn exclusively by men, it has become, in American English, a catch-all term for almost any garment other than outerwear such as sweaters, coats, jackets, or undergarments such as bras, vests or base layers...

, belt buckle

Belt buckle

A belt buckle is a buckle, a clasp for fastening two ends, as of straps or a belt, in which a device attached to one of the ends is fitted or coupled to the other. The word enters Middle English via Old French and the Latin buccula or "cheek-strap," as for a helmet...

, and trouser fly

Fly (clothing)

A fly on clothing is a covering over an opening join concealing the mechanism, such as a zip, velcro or buttons, used to join the opening. The term is most frequently applied to a short opening over the groin in trousers, shorts and other garments, which makes them easier to put on or take off and...

. In order to be properly dressed, these three should align to form a straight line down the front of a person’s body. The name derives from the use of the word "gig" to mean a violation or infraction during inspection. Unpolished brass, unshined shoes, and sloppy gig lines are all "gigs."

This stemmed back to the early Royal British Navy. A single-banked six-oared pulling boat, assigned to ships (usually for the Captain's use) in Her Majesty's Royal Navy back when they had wooden ships. Young officers with infractions during inspections were stuck cleaning the Captain's Gig, i.e. they were "gigged".