On the federal level, the U.S. government made it illegal in the 1800s to imprison someone just because they couldn't repay a debt. However, more and more often, courts in some states are jailing individuals who fail to repay a creditor when they are court ordered to do so.
The aforementioned practice is not widespread. Under the Fair Debt Collection Practice Act, debtors who do not pay civil debts (like to a bank or other company) should not be imprisoned, and creditors and debt collectors are specifically forbidden from threatening debtors with jail time. Nonetheless, willful failure to pay debts to the government (such as taxes) can land you in prison. It's worth noting that some of the signatories on the Declaration of Independence spent some time in now defunct debtor's prisons.