Contracts are agreements that create rights and obligations enforceable by law.1
When you think of a contract, you probably think about a document that people sign. Many contracts are formed this way. However, contracts do not need to be in writing. You can create a contract simply by talking with someone else. Most attorneys recommend against creating oral contracts because they are unlikely to protect you as well as a contract written by an attorney would. You also don't need to sign a contract in order to be bound by it. For example, two parties can create a contract by coming to an agreement over email.
Corporations rely on humans to enter into contracts. Before someone can enter into a contract on behalf of a corporation, he or she must have the authority to do so. In most startups, the board gives the CEO broad authority to enter into contracts, and empowers the CEO to then delegate that authority to others as they see fit.2 This is why contracts often ask for your title when you sign on behalf of a corporation. By specifying your title, you are indicating how you (hopefully) have authority to bind the corporation.