The statute of frauds requires a signed writing for particular categories of agreement. It is a defence to enforcement, not a rule that voids the contract, and it must ordinarily be pleaded or it is waived.
The classic categories are contracts for the sale of an interest in land; contracts not capable of performance within one year; promises to answer for the debt of another; contracts for the sale of goods above a threshold value; promises by an executor to pay estate debts personally; and, historically, promises made in consideration of marriage. Many states add credit agreements with financial institutions, which is why loan commitments are so carefully papered.
What satisfies it. A writing, not necessarily the contract itself, that identifies the parties and subject matter, states the essential terms, and is signed by the party to be charged. Multiple documents can be read together. Emails and electronic signatures qualify under electronic transactions statutes, which means a signature block at the foot of an email can matter more than the sender intended.
Exceptions. Part performance in land cases, particularly possession plus improvements. Specially manufactured goods. Admission in pleadings or testimony. Goods received and accepted, or payment made and accepted.
Between merchants, a written confirmation that is not objected to within ten days can bind the recipient under the UCC even though they signed nothing. That rule surprises people regularly.