Mulch is anything that is laid on the ground around plants in order to retain moisture in the soil and prevent weed seeds from seeing the sun. Mulch also helps keep the soil warmer, which is especially useful in the spring, fall, and winter.
Examples of mulch include landscaping fabric and plastic (usually black, but sometimes other colors; red is popular around peppers and tomatoes, since it warms the soil better than other colors). Organic mulches have the added benefit of feeding the soil and giving it nutrients as it decomposes. Examples of organic mulch include straw, wood chips, grass clippings, leaves, and yes, compost.
Compost is made from organic matter (such as vegetable and fruit leftovers, leaves, and paper products) that has decomposed. Finished compost looks like black or dark brown soil. It's usually tilled or dug into the soil (or used as a layer in lasagna gardening) in order to add nutrients to the dirt.
When is used as mulch, it may help block sunlight from weed seeds, but it doesn't do a very good job of retaining moisture in the soil. Also, just tossing compost on top of the soil, without working it in or covering it with some other type of mulch, means much of the nutrients in the compost aren't readily available to plants.
How to Mulch
Lay down your choice of mulch (I recommend organic mulch, since it feeds the soil and attracts worms who aerate the soil...and who poop, adding excellent nutrients to the soil) around plants, or on any bare soil. The mulch should not touch plant stems, or the stems become susceptible to rot and disease. The thicker the layer of mulch, the more it helps retain water and prevent weeds.
Sometimes mulch is also used to protect plants that are being overwintered. For example, in many places in the U.S., you can keep carrots, parsnips, and beets in the soil over winter. If you get snow, it's best to cover the crop with a thick layer of straw or other mulch - at least seven inches of it. The tops of the root crops will die, but the mulch prevents the edible root from going bad.
How to Compost
In essence, toss fruit and vegetable scraps, thin layers of grass clippings, thin layers of shredded grass, weeds (that haven't gone to seed), and paper products (large ones shredded) into a pile. Everything will decompose and turn into compost.
For details on the fastest ways to get compost, please read my post "Composting the Easy, Cheap Way."