Having an unlocked cell phone means your device is not locked to the cellular network carrier (e.g. T-Mobile) who originally sold the device and will work with SIM cards issued by other cellular network providers.
Usually, unlocked cell phones have the carrier's branding on the device. To test whether your phone is locked, insert a Telestial SIM Card or a SIM Card provided by a different carrier (than that of your device). If your device connects to a network and you are able to use the service, your phone is not locked. If your device does not connect to a network or you are unable to use the service, your phone may be locked.
Networks will claim that they lock phones to ensure customers see out their full contracts with them (most customers are paying off their cell phones over the life of a contract), however unlocking a phone for a different carrier has no effect on a service provider enforcing the full terms of a contract with a customer.
A carrier will lock your cell phone to their network as an effective barrier against a customer (like you) deciding to switch providers.
Refers to modifying restrictions on a phone to allow the installation of unauthorised apps or software.
'Root unlocking' or 'rooting'
Unlocking a phone's root administrator access so you can modify the operating system or install a different one.
Neither of these terms are relevant to using International SIM Cards.
To use Telestial SIM cards, your device will need to be GSM-compatible (Global System for Mobile Communications). In the United States, some carriers like Verizon and Sprint use CDMA 2G/3G networks, which is a different technology to GSM networks (which is used by just about every other country in the world). CDMA networks too are moving towards GSM technology, especially utilising the high-speed LTE networks.
To determine whether your device is compatible, please refer the device manufacturer's user guide. You may also check compatibility by determining if your device has a SIM Card slot as most CDMA-only devices do not use SIM Cards.