A process can be as simple as who next makes the coffee at the company coffee machine, or as complex as how a new product is made. Processes are a company's habits, the mechanisms that have built up over time for solving various problems. In a way, processes are like little computer programs. They have exact rules that can be written down. The difference is that processes are programs for a set of people rather than for a computer.
Processes can often differ from stated company policy.
Whenever things don't seem to happen as they should, and someone has to hand carry a transaction through multiple stages to expedite it, dysfunctional processes are probably at fault. Those stages should happen automatically without the need for an babysitter. For instance, it may be that an invoice arrives each month with payment due a month later. The reason official processes don't work is usually that they don't factor in real life. They are created by a person who hasn't consulted all the parties involved, or is in a different department than some of the people who will execute the process, and the resulting process just doesn't address reality.