Something that's paid is done in exchange for money, or happens while you're receiving pay. For example, your job might offer paid vacation, when you get your usual pay but don't have to work.
Paid work is anything you do for pay, whether it's babysitting your cousin for ten dollars, or a job at a law firm that provides a weekly paycheck. When people are described as paid, it clarifies the fact that they're working for money — like a paid FBI informant, or a paid companion for an elderly man. Paid comes from the verb pay, which meant "pacify or please" long before it came to mean "give what is due."