The radio frequencies used in US for cellphone communication are different from those in China and Europe. A cellphone in China or Europe cannot be used in US and vice versa, unless it supports multi frequencies.
A carrier is a company who provides you cellphone communication survice. Its service may be provided through dealers. For example, Kitty Wireless is one of the dealers for Page Plus, a carrier providing cheap and flexible service.
A set of service is called a plan. There are roughly three types of plans:
Pay-as-you-go is the cheapest if you don’t need the newest phones.