The most common for private individuals and smaller companies is to connect to the Web by phone, paid for as a local call for the time used: the solution is reasonably priced in these contexts, although comparatively slow.
Apart from the subscription to an 'Internet Service Provider' an electronic box is needed to 'MOdulate' the outgoing information-signal from computer-language to telephone-cable language, and 'DEModulate' the incoming one: thus the name 'modem'. In some countries you can buy one for a few pennies together with a subscription, or if you prefer to keep your choices open, you can get one independently for 1,000 times that.
A permanent connection is significantly faster, and becomes economically interesting for a larger organisation which uses the Web a lot: one example of many is JANet, the UK Joint Academic Network, to which many university teachers and researches are connected.
There are a number of other solutions too, of course