I received a 1099 because I was working for a political campaign. What does "Method of Accounting" mean? I never was involved in accounting and I was paid by check.

"Am I legally considered to be a business because I worked on a contract?" - yes, your business was providing campaign services.

"Why am I being asked to provide a business name and a business address?" - it's asking if you had a separate business name for your contracting. If you didn't, then don't enter anything. In your case, many people would not.

In your situation, your business address is probably the same as your home address.

"Method of Accounting" - refers to how you kept your books. You did billing and paid expenses, right? Therefore you had "books". In your case, most people just answer "cash".