Typically when attaching a camera to a tripod it is designed to be attached in the landscape position via the threads on the bottom of the camera. This potentially creates three problems for the photographer if/when the photographer wants to rotate the camera into the portrait orientation:

1. Once rotated, the weight of the camera is no longer directly over the center of the tripod, potentially compromising the stability of the tripod (especially with heavier camera/lens combinations)
2. Once rotated into the portrait orientation the photographer must recompose their shot as the camera has been shifted down and to the left of where it was initially positioned.
3. On some 3-axis tripods, the handles get in the way such that it is difficult or impossible to completely rotate a camera into the portrait orientation.

An L Plate is a... well... 'L' shaped quick release plate- often milled out of aluminum- that attaches to the bottom of a camera similarly to any other tripod quick release plate. The difference is that it extends across the bottom of the camera and up the side, as visualized in this diagram.

At its essence it is a device that allows the photographer to rotate the camera from landscape to portrait orientation by simply detaching the quick release plate on the bottom and attaching the camera again via a second quick-release plate positioned on the side of the L plate and camera. This has the advantage of keeping the camera positioned squarely over the center of the tripod (thus ensuring stability and vibration dampening regardless of camera orientation, as well as ensuring that tripod handles aren't extended such that they interfere with the rest of the tripod), and meaning that the photographer doesn't have to recompose his shot once the camera has been rotated, because the camera hasn't been shifted down and to the left.

While there are several companies that make 'L Bracket' products, and there are various tutorials available to roll your own, the phrase and the product 'L Plate' is most commonly associated specifically with products manufactured by the company Really Right Stuff.