Valid contracts can take different forms. The type that as a business professional you may be most familiar with is known as an express contract, which is an agreement set forth in a signed writing and the terms of which are included in the agreement document.
There are a couple of other contract types that you may not be familiar with, but which are still enforceable. One is the oral agreement, in which the two sides agree to a course of performance without ever setting forth the details in writing. The other is an implied contract that a court may impose on the parties, which is also known as a "quasi contract."