Nothing stops a landlord from renting to you under any circumstance, even when you owe money on a previous apartment. For practical reasons, a landlord probably wouldn't rent to you with a delinquent lease on your credit history, because you may break your new lease, too. Landlords often require at least an average credit rating --- about a 620 --- and no broken leases; the ideal applicant should have a rating of at least 670.
Not all landlords require a credit check. Owners of large apartment complexes usually run a credit check on applicants, but landlords of smaller complexes may not run a credit check due to the cost. Mom-and-pop complexes may allow more flexibility as far as their credit rating requirements. You may have a chance to explain the delinquent rent, such as a period of unemployment causing you to miss rent and that you paid off what you owe to your previous landlord.
Paying Off a Broken Lease
Landlords usually don't report to the credit reporting agencies, so delinquent rent probably appears on your credit history because your previous landlord sued you in small-claims court or sold the debt to a collection agency. Talk to your previous landlord about settling the debt or paying it off in installments. A paid collection or judgment doesn't improve your credit rating, but it usually looks better than ignoring the debt.
Start repairing your credit history before you look for a new apartment. Bankruptcies and frequent late payments are red flags to a landlord, so make it a priority to pay bills on time. You may have to offer the landlord a larger security deposit than that required from a normal tenant, as well as let the landlord deduct rent directly from your bank account. You may also offer to prepay several months of rent or the entire lease.