The best mulch for a vegetable garden include organic, biodegradable materials like untreated grass clippings, clean straw, or leaves – as these all help to suppress weeds, retain moisture, and also break down to enrich the soil life which then feeds your plants.
IMPORTANT TO NOTE:
Keep mulch 1-2 inches away from around the base your plants to keep them from rotting. Piling mulch against the stems of your vegetables can trap moisture and create an environment that encourages fungal diseases, stem rot, and attracts pests—essentially suffocating the base of the plant. Same advice applies to mulching your trees and shrubs, if you use wood mulch, be sure not to create volcano mulch piles around their trunks!
Where to mulch: Mulch is especially helpful in dry, hot growing regions like ours in the West. If you live in a rainier region, mulch may not be ideal if the soil is already consistently moist, as it can trap excessive dampness and foster rot or pests like slugs.
Mulching pots is a great idea!
You can use mulch in vegetable pots, it helps greatly conserve moisture and slow down the water from draining through as quickly, which helps retain more nutrients. 
Grass Clippings:
Grass clippings are a free, nutrient-rich mulch for vegetable gardens! If you use grass clippings, be sure that they are untreated (no herbicides or pesticides used), and ideally it's best to let them dry out a bit before putting on your garden beds so they don't turn into a slimy mess. You can spread fresher grass clippings in thin, 1-inch layers, allowing them to dry out before adding more. You'll find that over a few weeks they start to disappear as the soil life consumes them, so keep topping them off throughout the summer to keep your soil and veggies happy.
Covering the soil protects the soil life from UV rays, while conserving moisture and regulating temperature, keeping your vegetable garden happier!
On super hot 90˚ F + days, vegetables can wilt to conserve moisture. Keeping them mulched helps keep the soil temperatures cooler, reducing evaporation and conserving moisture!
Leaf Mulch & Leaf Mold:
Don't throw away your trees' autumn leaves! This free resource is invaluable for your veggie garden! Pile leaves onto your vegetable or flower beds during the winter to insulate the soil while conserving moisture. In the spring many of the leaves will have broken down. The ones have haven't can be used to mulch your vegetable garden. Some vegetable gardeners even ask neighbors for their fall leaves if they are not reusing this vital resource, so don't be afraid to ask if they are throwing them away!
What is leaf mold?
Leaves naturally break down over time into leaf mold – which is a dark, crumbly, and earthy-smelling material created entirely from decomposed leaves. Leaf mold can absorb up to 500% of its own weight in water, helping sandy soils retain hydration and preventing drought stress in plants. Leaf mold lightens heavy clay soils and adds spongy organic matter to aerate the earth for strong root growth. Leaves naturally attract earthworms and beneficial microbes to your vegetable and flower beds so they are one of the best free resources to use for your vegetable garden! 
Plant Pruning Mulch:
Oregano and other herbs or perennial prunings can also make great mulch! If you have too much oregano or tarragon (above), for example, you can use them to mulch your vegetable garden. Using excess oregano, tarragon, or other herb prunings as mulch is a great technique. Because of herbs' strong aromatic oils, these clippings not only help retain moisture and regulate soil temperature, but they also act as a natural pest deterrent for your vegetables.
All of these organic mulch types can help to deprive weed seeds of light, which help reduce weeds in your veggie garden. Be sure to re-apply mulch over the summer months if it starts to disappear (grass clippings and leaf mulch especially will slowly get broken down by the soil life) so reapplying is essential to continue to enjoy the benefits of mulch. Of course, always be sure to leave room around the stems of your plants so you don't smother them. 
Be sure to mulch your garden beds (especially if you live in a dry growing region), and reap the rewards of happier soil and plentiful harvests!
