You buy a SIM card from a wireless provider, register your service, and you're connected to that provider's network. Depending on which country you live in, you may be asked to provide evidence of your identity to varying degrees.
For example, to set up my service in Australia, I bought a SIM card in a little packet at a corner store for A$40. The SIM card was from a specific wireless provider (Telstra, in this instance), but it was sitting side-by-side with SIM cards from other providers in the store. It came "pre-loaded" with 35GB of data to use on their network, plus unlimited SMS and voice calling, which was described on the packaging. When I put the SIM in the device I was prompted to go to a website to register my service, where I input a code from the SIM card packet to link the SIM itself. I was asked to provide my driver license number or passport number. I was then able to choose between three pre-selected phone numbers that were displayed on the screen. Click Finish, and my cellular service started working on my phone. Every month I have the option to renew my plan under the same terms, or choose a plan with different terms (more or less GB of data, for instance), or throw the SIM card away and switch to a different wireless provider.
If we ignore prepaid options it really is identical to your current bundle of phone contract plus smartphone loan - minus the loan. You sign the contract, get sent a SIM card and pop it in the phone you bought elsewhere and off you go.
Actually in Europe for example, most prepaid sim cards also offer setting details through a web portal, and it just automatically takes money off your card on a set date in a month to recharge the SIM.
