In legal terms, a contract is any agreement between parties to exchange things of value, such as goods and services for cash. A simple handshake agreement is legally binding, but its terms are not always provable if a dispute arises. However, you don't necessarily need pages upon pages of legalese. A basic contract agreement often works best for a simple business transaction.
If you run a business as a sole proprietorship, it would be appropriate to be identified in the contract as John Jones doing business as Jones Plumbing. If you organized your business as a Limited Liability Company, identifying your participation in the contract by your name could remove any personal liability protection that an LLC provides. Similar issues can arise with a partnership if each individual is identified as a party to the contract.