Your rights depend on whether it's a domestic or international flight.
Domestic flights. For the most part, federal law does not require airlines to compensate passengers whose flights are delayed or canceled. So any compensation you'll get will depend on the airline's policy. Full-service airlines are likely to offer more generous terms, such as meals, hotels, alternate transportation, or even emergency toiletries in the event of an overnight delay, while budget or no-frills airlines may offer little, if any, compensation.
International flights. If the delay was caused by anything other than the airline's overbooking, recovering damages for an international flight delay can be very difficult. Under an international treaty called the Warsaw Convention, an airline can escape liability for damages caused by flight delay if it can show that it took all necessary measures to avoid the damage or that it was impossible to take such measures. If your international flight is delayed, you may be able to persuade the airline that it should cover direct costs caused by the delay, such as meal, hotel, or telephone expenses. To back up your argument, you can quote Article 19 of the Warsaw Convention which states: "The Carrier shall be liable for damages occasioned by delay in the transportation by air of passengers, baggage or goods."