Cover crops offer the same advantages to backyard home gardeners with raised beds as they do to farmers with large acreage, and are a major component of the Three C's of organic gardening (compost, crop rotation, and cover crops).
Unlike harvestable crops, a cover crop (aka green manure) is a plant grown to help revitalize the soil and replace nutrients that vegetable crops inevitably remove from the soil during the growing process. It may seem a bit strange to grow a crop only to dig it back into the soil, but that's how cover crops work. Cover crops improve soil quality, increase porosity (aeration), reduce compaction, add organic matter and beneficial microbes, retain nutrients, add nitrogen, suppress weeds, attract beneficial insects (certain cover crops), and generally enhances overall soil biodiversity.
There are so many soil types and so many gardening zones across the country that there's no way to designate a 'best' cover crop for backyard vegetable gardens. All cover crops offer benefits, but certain cover crops provide specific benefits such as breaking up compacted soils, fixing nitrogen, or attracting beneficial insects. Some are winter cover crops, while others are summer cover crops. Our soil is sandy so we don't need a cover crop that helps break up compacted soil, but we do need a crop that helps prevent nutrient leaching, adds organic matter, retains moisture, and attracts beneficial insects. And since we're in the Northeast, it's helpful to have a cold-hardy cover crop for early spring planting. The three crops we've used so far are red clover, peas, oats, vetch mix, and buckwheat.
There are no hard and fast rules for when to turn cover crops into the soil but the longer they're allowed to grow, the more difficult it will be to turn the cover crops under. It's not only more work, but the increased digging will inevitably lead to more earthworms getting chopped up in the process. Peas, oats, and vetch tend to create a thick carpet of vegetation, while buckwheat is more of an individual plant which makes it easier to dig into the soil. If the cover crop gets too large you can always pull up the plants by hand or cut the crop with a string trimmer, then either turn the pieces into the soil or add them to the compost pile. The images below show the cover crops being turned into the soil prior to planting corn. Half of the peas/vetch mix was pulled by hand and the rest turned into the soil.
It's tough to figure out how much cover crop seed you might need, and I can only offer what works for our application. My raised beds are different sizes but are roughly equivalent to fourteen 4' x 8' beds. We sow the peas/vetch mix in every bed in the springtime and use the buckwheat throughout the growing season when crop rotation allows. In February 2015 we bought five pounds of buckwheat and five pounds of the peas, oats, vetch mix at $20 per five pound bag. By the end of the 2016 season the peas/vetch mix will probably be used up, but there's plenty of buckwheat left for the 2017 season. They also sell an inoculant for the cover crops which was used the first season, but the inoculant has to be purchased new every year so we haven't used it again since the first season.
There are several places that sell cover crop seed but we usually buy our seed from High Mowing Seeds or Johnny's Seeds. Either one would be a good place to start in order to get a list of the various cover crops along with their growing characteristics. If you have a small garden and don't need much seed, Johnny's offers cover crop seed in smaller amounts than most places. When searching for a particular seed such as organic buckwheat, you may need to include 'cover crop' in the search term or you may get sprouting seeds in the search results instead of cover crop seed.
The first video looks at the peas, oats, and vetch in one raised bed, and a buckwheat cover crop in another raised bed that were in place prior to planting a corn crop, and the second video is a buckwheat cover crop in full bloom with scores of insects buzzing frantically amongst the flowers.