Garden vegetables belong to families of related plants. These families tend to share pests and diseases.
You don't want last year's pests lying in wait for the new seedlings, or diseases in the soil around your seeds. Moving each crop family to a new spot from last year helps give them a good start. For best insurance against bad diseases like soil-borne wilt, allow 4 years before the same family returns to the same place.
Pests and disease spores overwinter in the soil and on crop residues. You can address soil-borne problems with rotation. Get rid of diseases or pests lodged in crop stems, leaves, and other residue by hot-composting all crop residues if the fall.
Making it Work
If you have a number of similar-size beds, rotation is easy. Just divide the crops up into beds, and then always move things one bed over each year. You can use this system to help with the fertility needs as well: lots of manure or compost and lime on the beds with heavy feeders this year, and the next year a light feeder like carrots can move into that space with no additional amendment.
A time-honored system for a single garden space is to divide it into 4 or 5 roughly equal areas, and rotate each family (or group of families) into a different one every year until at last they get back to where they started.
You would need to group families that don't take up much space, or that you don't grow much of, together. A family like tomatoes/potatoes/peppers, would fill an area on their own. Area 2 might be squashes, corn, and perhaps beans (the "three sisters") Area 3 might be onions and brassicas, which both like lime and nitrogen. I put these in one area for spring and a different area for fall. That would leave a final area for everything else--lettuce, spinach, carrots, beets, chard, okra, etc. If you like to grow lots of corn, or lots of winter squash for storage, each of those might fill an entire area.
Some plants have worse potential for disease; those are the ones you should make sure to rotate. If you get onion thrips, for example, be sure that the onions move in a group and don't return until year 3 or 4. Other plants that have fewer problems in your area might fill in wherever there is space.
Generally speaking, brassicas (cabbage family) have more possible diseases and pests than other families, so in most parts of the country you'd do well to rotate them.
The other extreme is to use polycultures of many types of plants mixed together in each bed, and trust that the mixture will deter pests.
My own system is to rotate the families with major pest or disease potential here--the tomato, squash, spinach, and cabbage groups. I scatter the rest where space allows, using a lot of companion plants like flowers and herbs.
Here is a gardener's list of families:
PARSLEY FAMILY (Umbellifers): carrots, dill, fennel, parsley, parsnips, cilantro, caraway, celery,
ASTER FAMILY (Composites): lettuce, chicory, sunflowers, artichokes, cardoon, endive
CABBAGE FAMILY (Brassicas): broccoli, cabbage, cauliflower, mustard, Asian greens, Brussels sprouts, Turnips, radishes, cress.
TOMATO FAMILY (nightshades): tomatoes, potatoes, peppers, eggplants, tobacco, petunias
SQUASH FAMILY (cucurbits): winter and summer squash, cucumbers, melons, watermelon
BEAN FAMILY (Legumes): peas, beans, cowpeas, vetch, favas, soybeans, astragalus, sweetpeas
SPINACH FAMILY (goosefoot or chenopods): spinach, chard, beet, quinoa
HIBISCUS FAMILY (mallows): Okra, hollyhocks, marshmallow, hibiscus, jamaica, Egyptian spinach
GRASS FAMILY Corn, wheat, rye, bamboo, barley, rice, sorghum, millet, teff
ONION FAMILY onions, leeks, lilies, chives