To connect to the internet you're computer needs to have a connection to a network that is on the internet. Most internet users connect to an Internet Service Provider through the telephone system using a Modem. The ISP has a large set of modems waiting for incoming calls from its users. Once a connection has been made the user's computer can behave as if it were directly on the internet. Other systems of connecting to the internet exist, such as cable modem access and even satellite connections.
Most users choose telephone modems because of the low cost.
If cost isn't a question then you can build your own network and connect it to the internet using a dedicated connection. With a dedicated network connection you can offer services from your network to the world, such as hosting your own web server. Since every computer on the internet speaks the same basic language, they can all exchange information. Modems are hardware devices that convert network traffic into a language of sounds that can by transmitted over the analog telephone system. This is a multi-vendor standard, and will be implemented by all modem manufacturers.