You've already had some good answers but I'll chuck in my tuppence worth too, if I may:
To use your phone out in the street (away from your home or any free wi-fi services, such as those now found in many pubs, cafes and supermarkets), it has to be able to access signals from mobile phone masts. To do so, you need to be signed up to (and pay money to) a service provider. i.e. such coverage is NOT included with your BT broadband package.
There are four companies that operate mobile phone masts in the UK. (Vodafone, O2, EE and Three). However each of them allows other companies to run services through its masts. (e.g. if you sign up with Asda Mobile, your phone will connect through Vodafone's masts. If you go with Tesco Mobile, it'll be O2 that's actually providing the service). It's often much cheaper to use one of those 'piggy-backing' services, such as Asda Mobile, Tesco Mobile or GiffGaff, than it is to pay directly to the companies who actually operate the masts.
There are three different ways to sign up to a mobile phone service provider . .
Firstly, you can have a contract. Just like a broadband contract, that ties you into paying a fixed amount for, typically, a period of two years. (Unless you're looking to find a way to get a mega-expensive phone that effectively comes on credit in the same package as your call charges do, I wouldn't recommend going with a contract deal).
Secondly, there are 'conventional' pay-as-you-go (PAYG) deals. A good analogy here is a pre-payment meter for electricity, where you pay so much in advance for what you think you'll need but can then top it up later on if you need more credit. It's just the same with PAYG phones; you just 'top up' the phone with credit from time to time, so that your calls are paid for in advance. The advantage of such a way of doing things is that you're not tied into a contract. The big disadvantage of the system though is that it usually costs a LOT more to make calls and use data (for internet access, etc). So I don't recommend going down that route either.
The way the majority of people seem to pay for their mobile phone usage these days is to use 'bundles'. (That's the term that most phone service providers use but GiffGaff calls them 'goodybags'). Using 'bundles' is a way of getting PAYG freedom (i.e. there's no contract involved) but paying really low prices for calls and data at the same time. You simply buy a 'bundle' each month that gives you a certain number of minutes calling time, a certain number of texts and a certain amount of data for use in that month only. (i.e. unused credit isn't carried over to the next month, except in some O2 deals) In many cases the calling time and data can actually be unlimited. Most people set things up so that their 'bundle' payments are taken from their debit cards automatically every month but, as there's no contract, they're free to switch to a bigger or smaller monthly 'bundle' at any time (or even to simply cease using that provider's services).
The most popular service providers among AB members seem to be Asda Mobile, GiifGaff and Tesco Mobile. I'll use Asda Mobile here as an example, simply because that who I'm with:
https://mobile.asda.com/bundles
As you can see from that link, you can get unlimited calls and texts for a fiver a month, together with 3GB of data. That's enough for a bit of occasional web browsing (such as checking the weather or train times) but insufficient if, say, you'll be watching loads of YouTube videos. I actually pay a tenner a month for 12GB of data but I've never actually used anywhere near that much. (So I could actually drop down to just £7, or even £5, a month, if I chose to do so).
Moving onto actual smartphones now . . .
Some phones are sold 'tied' to a specific network. That means that you're stuck with paying the (often high) prices of whichever company whose services you're forced to use.