American cellphone carriers

Calling all Americans,

Can someone please explain to me, fairly simply, how mobile networks work in the US.
coming from the UK i am pretty naive and ignorant to how they work in the US.

I know a little, for example LTE, 3G, etc but then theres things that i am not sure on. The same phone that only works with certain carriers ie a Verizon or an AT&T phone. T-mobile not working with anything? Whats AT&T 4G is it actual 4G. infact what is 4G as far as i know we only have 3G in the UK.

I'd love to learn a little more so i understand a lot of what the guys at the verge mean when they refer to certain devices and networks.

Cheers, Tom