Wireless carriers - also called wireless service providers or mobile network operators (MNOs) - are the companies that operate the wireless networks and sell use of those networks (the service).
The network consists of antennas linked to base stations (commonly called "towers"), and infrasructure linking them. The service is simply allowing customers to access the network, typically billed by the minute.
Some carriers offer flat-rate plans. Still others, called MVNOs, do not own a network.
"Carrier" can also refer generally to the radio waves "carrying" the signal, or specifically to a small part of a frequency band.