An unlocked cell phone is one that is not tied into a certain carrier's network: It will work with more than one service provider.
Most cell phones and smartphones are tied—or locked—to a certain cellular carrier, such as Verizon Wireless, T-Mobile, AT&T, or Sprint. Even if you don't actually buy the phone from the carrier, the phone is still tied to a carrier. For example, you could purchase an iPhone from Best Buy, but it still requires you to sign up for service from AT&T.
For many people, purchasing a locked phone makes sense: The carrier offers a discount on the handset in exchange for you signing a service contract with them. And, in addition to the discount, you also get the voice and data service that you need to use the phone.
But not everyone wants to be tied to a certain carrier’s network, for a variety of reasons. If you frequently traveloverseas, it may not make sense to be tied to a phone that won't work internationally (or one that will cost you an arm and a leg to use in foreign countries), for example. Other people are unwilling to sign the lengthy service contracts (two years, typically) that many carriers require. That's why purchasing an unlocked cell phone or smartphone can be a desirable alternative.