In the early days of broadband, the network was mainly run through the BT infrastructure. This is still the case for the most part, but in many ways the market and the networking options have opened up a lot. Phone-line broadband is also known as ADSL.
Depending on your needs there are a range of ISPs to choose from, some may use the BT ADSL network while others don’t. This largely depends on your choice of service (broadband, mobile broadband, fibre, or cable).
For the most part, ISPs use the BT ADSL network, so even if you choose for a different provider there will be some line rental to pay to BT (which goes through the normal pricing, so you won’t have 2 bills to pay). There are other networks, notably TalkTalk, as well as some more niche independent operators mainly focused on fibre optic broadband (such as thinkbroadband). There are even some villages that have laid their own superfast broadband networking!
But mostly if you aren’t opting for mobile, fibre optic or cable broadband, the ISP you choose will probably run their services on the BT network. This is not too much of a problem, since BT are contracted to provide the same high level of service they provide for their own customers.
Cable and fibre optic is a different story. Virgin Media (who bought the cable network of NTL Telelwest group) have their own cable broadband offering (with high speeds), and there are several niche fibre optic broadband providers for home and business that can also provide very high speeds on their own fibre network.