A classic example of companion planting is the Native American "Three Sisters," or corn, beans and squash. The beans enhance nitrogen levels in the soil; the corn provides a living trellis for beans to climb; and squash creates a living mulch that minimizes weeds in the vegetable bed.
The easiest way to companion-plant for weed control is to sow a cover crop, such as clover, between rows of plants. Grasses such as clovers can assist in maintaining nutrient content in the soil and minimizing unwanted weeds, and are compatible with a wide variety of plant species.
Pairing vegetables so that one vegetable species provides weed control for another requires a higher degree of planning. In the "Three Sisters" example, small amounts of the chemical juglone, best known as the chemical that gives black walnut its toxicity, may be responsible for squash's high success in weed suppression. Corn and beans are both tolerant of juglone and thus pair well with squash. Companion-planting guides can be found on the Internet, from which you can determine which plants pair well together.