A full plaid is a long piece of tartan

Tartan

Tartan is a pattern consisting of criss-crossed horizontal and vertical bands in multiple colours. Tartans originated in woven wool, but now they are made in many other materials. Tartan is particularly associated with Scotland. Scottish kilts almost always have tartan patterns...

 fabric, traditionally worn as part of a full highland dress uniform. It usually matches the tartan of the kilt

Kilt

The kilt is a knee-length garment with pleats at the rear, originating in the traditional dress of men and boys in the Scottish Highlands of the 16th century. Since the 19th century it has become associated with the wider culture of Scotland in general, or with Celtic heritage even more broadly...

. A full plaid is pleat

Pleat

A pleat is a type of fold formed by doubling fabric back upon itself and securing it in place. It is commonly used in clothing and upholstery to gather a wide piece of fabric to a narrower circumference....

ed the whole way, with half of its length sewn shut (so that the pleats cannot open). Its length is almost twice the height of the wearer (about twice the distance from the ground to the wearer's shoulder).

A full plaid is typically only seen on members of pipe bands which elect to wear full dress (military styled) uniforms, and occasionally as formal highland dress.

The full plaid is wrapped around the wearer's chest and under right arm, pulled firm to the body. The plaid is twisted on the left shoulder with one loose end falling behind the wearer's back and getting tucked into the waist belt. The leading edge of the other loose end is pulled forward and draped over the wearer's left shoulder.

Properly worn, the edge of the plaid should be level with top of left spat point at the rear of the leg, with fringe hanging down below this level. The lower edge of the plaid should be horizontal and parallel with the ground. The leading edge and front face of the plaid is secured by a brooch (often a cairngorm brooch) on the left shoulder.

To the lay person, it would appear as a sort of long, tartan, shoulder-cape.