A "body" is a figurative way to say "organization", or "institution".
A "professional body" might be a company or corporation, some kind of agency, or various other privately-run entities with a wide range of purposes. For example, a real-estate agent might have to pay a fee to take a test offered by a local licensing board, or pay yearly dues (a kind of fee) to a representative agency. These entitles are responsible for tracking and training all authorized real-estate agents in that area, or might advocate with the government about laws related to real estate.
Meanwhile a "public body" is usually a government-run entity responsible for administering some particular field. For example, in the United States, each state has its own Department of Motor Vehicles responsible for licensing drivers and registering vehicles. If you own or drive a vehicle, you must pay regular fees for the renewal of official documents.