Cellphones use a SIM card to identify a handset on a particular network, and different cellphone carriers have different network frequencies. When travelling abroad, especially in places where you may transfer between different cellular carriers' operating regions, dual SIM card phones are very convenient. The phones do work just fine with a single SIM card.
Early Dual SIM Card Phones
The earliest dual SIM card phones had one SIM card slot and a socket adaptor that would let you put two SIM cards into a protective sleeve and then use both; these also had some custom software that allowed you to switch which SIM card to use. Some of the very earliest of these devices -- circa 2005 or so -- required that you solder the adaptor to the phone. This type of phone works just fine with a single SIM card.
Dual SIM Socket Phones
More advanced designs have two SIM sockets in the body of the phone itself. They also have a few additional user interface controls, like two call buttons. These phones can run with either one or two SIM cards, though if you have only a single SIM card, only one of the call buttons will work. This type of phone typically has two transceivers or antennas; if you have a single SIM card in the wrong SIM slot -- you may not be able to connect to your cellphone carrier at all, so make sure you've got, for example, the AT&T SIM card in the slot connected to the transceiver that uses the AT&T antenna.
Benefits of Dual SIM Phones
One of the major benefits of dual SIM phones is convenience during international travel. The other place where they represent a great deal of convenience is when a cellphone user is obligated to carry a work-specific cellphone and has a personal cellphone. A dual SIM phone lets them put the SIM cards in for both, be able to identify which type of call they're taking by looking at the incoming caller ID, and answer appropriately -- and without having to juggle two or more phones.
Drawbacks of Dual SIM Phones
Dual SIM phones, often using two different transceivers, typically have shorter battery life because they're running two different radios and checking for cellular signal with each of them. Some models have larger batteries or two batteries to compensate for this, and to date, only a handful of smartphones have dual SIMs due to battery life issues.