Generally in the U.S., states have laws that require some contracts to be signed by a party in order to enforce the contract against the party. These requirements are generally set out in a law commonly known as the Statute of Frauds. See the Wikipedia article. Although the exact details will vary from state to state, the following types of contracts generally would be unenforceable without a signature: contracts in consideration of marriage (e.g., prenuptials, postnuptials); contracts that cannot be performed within a year; real estate contracts; contracts for goods above a particular threshold, such as $500; and surety contracts.
Even if a contract does not require a signature, it is generally wise to do in order to prove the parties agreed on the terms of the contract. And if the contract is being submitted into evidence at trial, a notarized signature could eliminate claims against its authenticity. See, e.g., Federal Rules of Evidence Rule 902.
If the contract is not signed (or even not in writing), generally a witness would have to testify that the parties agreed to the contract. This of course can devolve into a he-said, she-said argument and the judge or jury would have to decide who is right.