In the US there are two type of cellular network, GSM (which is more of a universal standard) and CDMA. We have four major carriers, AT&T and T-Mobile which are GSM. Sprint and Verizon which are CDMA.
They aren't interchangeable, so you cannot use a GSM phone on a CDMA network. (Some carriers support both CDMA and GSM, but they generally are prepay carriers that lease cell towers from bigger network.) Also new Verizon phones have the hardware to support both GSM and CDMA, however, Verizon network itself only use CDMA.
When you get a cellphone plan in the US, 'generally' you sign a post-paid (pay at the end of the month) two year contract, by signing the contract you get to buy the sim/network-locked phone from the carrier for subsidized (or discount) price, so a flagship phone can be as low as $199 instead of $600. You don't actually own that phone until the end of the 2 years period. If you stop paying or switch to another carrier (ie. port over your number), you will be forced to pay an ETF (early termination fee), the carrier may also blacklist your phone IMEI number so it can't be activate (I'm not sure under what the specific condition they do that since it never happened to me). They may also require a credit check before opening a plan for you.
We can also buy subsidized phones from third party retailers that provide further discount (2 years contract required), such as Amazon or RatShack. Amazon and RatShack have their own ETF fee ontop of the carrier's ETF (to cover the cost of the phone), so you'll have to pay double if you terminate contract, but you have a time period for refund.
After the 2 years contract period you get to own the subsidized phone, then you can renew the contract for another 2 year and get another subsidized phone. Under special conditions the carrier may allow you to upgrade subsidized phones sooner than the 2 year mark.
T-Mobile is one carrier that do not do the 2 years contract anymore. You can sign up for their post paid plan, but you can leave anytime without paying ETF fee. You don't get subsidized price, but you make an initial deposit for a phone that you choose, then pay monthly for it on top of your regular plan until you pay off the full cost of the phone. So for example, if you pay $50/month for the plan, you pay another $15 per month for the phone. Once you pay off the cost of the phone, your total monthly bill will be $50 only and you own that phone. This is in contrast to the subsidized plan, where your monthly bill will never decrease. But T-Mobile charge a little more money for their phones, so its better to just buy an unlocked phone elsewhere, then create a plan with T-Mobile.
Other than this, there are lots of prepay option as well, where you buy your own phone and not tied to any contract. Generally post paid plans have more benefits because you get more support and features, such as '411' and roaming, which may not be available for prepay options.