Nodules on a fava bean root
Plant life is just incredible. Some plants actually feed the soil while they grow. A cover crop, also called a "green manure", is a plant that fixes nitrogen in the soil. Actually, the plant doesn't fix the nitrogen, but bacteria which live in a symbiotic relationship with the roots do this. Nitrogen exists as a gas in the atmosphere, which is unavailable to plants. Some plants are able to absorb nitrogen from the air and store it in their roots on nodules. The nodules are white bulbous growths on the roots. The presence of nodules doesn't always mean fixation--cut them open, and if they're pink, they have fixed nitrogen. Plants in the legume family, notably beans, peas, and clover, and also weeds like vetch will host nitrogen fixing bacteria. There may be other kinds of plants which do this too--some grasses like oats, may also help fix nitrogen. The bacteria usually are already present in the soil. But to be sure of their presence, you can first innoculate the seeds with the appropriate bacteria, available from a mail order farm supply store. The way cover crops work, you grow the plants from seed (innoculated or not), and when the plants have grown and are 50% in flower, you dig them into the soil. For less soil disturbance, you can simply cut them down, and mulch them in place. (Note: favas will resprout if the roots are not dug up)
For this reason, you probably don't want to grow super delicious snap peas to fix nitrogen--because you would destroy the plants before the peas form. My understanding is it's an "either/or" situation--either you eat the crop, or you dig it in to fix nitrogen. For this reason, most cover crops are things people don't tend to eat--some nurseries or seed companies sell a mix of bell beans, vetch, and some kind of oatstraw--all "weedy" things that grow well, but can be turned under without any regret. Fava beans are dual purpose, so you could grow some to eat, and turn under some also.
Farmers turn under the crop with a disk harrow (plough takes it too deep to do any good), but the small scale grower can just use a shovel. I cut the favas first , and then dig them in to the soil about 6 inches or less. (Children may enjoy doing this). There will still be some pieces above the ground, but the roots should be thoroughly disturbed, and most of the greenery buried. Then I cover the area with rice straw (I buy a bale from the local horse track, and let the chickens use it first), about 2 or 3 inches thick. It has to sit at least a month before you can plant into it. In a month, you won't see any greens, only dark rich soil.
Don't forget to water it in the meantime, as this enables the microbes to break it down!