In a nutshell, organic matter is various forms of living or dead plant and animal material. Compost is the prime example of this.
Non meat eating animal manures are also a good source of organic matter.
Mulches break down to form organic matter. It is not a factory produced fertilizer. Organic matter adds nutrients to the soil. It facilitates drainage as well as storing water and nutrients. It helps increase air drainage as well as the number of valuable microorganisms.