If you've bought an unlocked smartphone, you'll need a SIM (subscriber identity module) to get service. A SIM, sometimes called a SIM card, is a small card you slide into the phone (usually near the battery), that provides the phone with its phone number, as well as its voice and data service. Only GSM carriers--the two biggest in the U.S. are AT&T and T-Mobile--use SIMs.
Buying a SIM can be tricky, though. Some carriers sell them, but may require you to commit to a service plan on the SIM--which doesn't make sense, especially if you bought an unlocked phone to avoid this type of commitment in the first place.
You can also find prepaid SIMs from some third-party sellers. These can be a great idea, especially if you're planning to travel internationally. You could, for example, buy a SIM with a phone number local to the country you'll be visiting. This allows you to make local calls while you're there, instead of pricey international calls.
Purchasing and using unlocked cell phones and smartphones is becoming more popular, and for good reason. It can give you more freedom to use your phone as you like, and it can save you money in the long run. But finding the right phone and the right SIM to use with it can be confusing. Take your time and do you research before making a purchase. Good luck!