Lasagna gardening is simply a method of building a raised bed. Several different materials are layered over soil, grass or even concrete, creating what is essentially a large compost pile. The "garden" is either allowed to break down over several months or can be planted in right away. Mulch is layered on top now and then to retain moisture and restore nutrients. That's it!
Here is the lasagna "recipe": First, lay down a thick layer of wet newspaper or cardboard boxes to smother the existing planting (usually grass). Then alternate two-inch layers of peat moss with six-inch layers of organic matter until the bed is one and a half to two feet tall. The bed will get shorter as the materials decompose.
This is a good way to quickly start a new garden bed, assuming you can obtain all that organic material at once. The Square Foot Gardening method of building a raised bed is very similar, using weed cloth instead of newspaper to smother grass, and a mix of vermiculite, peat moss and compost (or straight compost) is laid down instead of layers of materials that will turn into compost. The square foot garden method sounds a little easier, the lasagna method could be cheaper; you choose!