Mulching is covering the soil with a thin layer of material to block weeds, retain moisture, increase nutrients and microbes in the soil and/or for appearance sake. There are different reasons to use mulch and different types of mulches.
Ground bark mulches are good spread around trees and shrubs. Ground bark mulches will suck nutrients away from plants so I don't recommend using them near flowers and vegetables.
Mature stable compost is better around flowers and vegetables because it will add nutrients to the soil not take away.
Shredded leaves from the year before are great to use as a mulch. Unshredded leaves may become matted and repel water.
Straw can be used but I find it to be expensive. Hay will bring lots of unwanted weed seeds into your gardens. Maybe if the hay is well rotted it would be okay. I'm not sure but that sounds like fun to find out on a trial spot.
Newspaper and cardboard are very good to block weeds and retain moisture. These covered with a layer of compost is my preferred mulch. They need to have something on top of them to weigh them down or they'll blow away. Without a layer of compost, shredded leaves, bark mulch, and etc. they are rather ugly.
There are also man-made mulches that may fit your needs. Black plastic and landscape fabric will block weeds. One problem with black plastic is in the heat of summer it gets so hot underneath the poor little microbes die off. You can also use free mulches which may include mud flaps, old rugs, and carpeting or use your imagination. Special attention should be made to make sure that you're not adding unwanted harmful chemicals from ingenious mulches.
The main reason I use mulch is to lessen the amount of weeding I have to do. Weeding never stops but by covering the rows and unused areas a lot of area can be eliminated.