To connect from home you need several things. Apart from a computer (obviously!) you'll need a phone connection, a modem or ADSL router, and some Internet software. Things will be easier if you're using a relatively recent operation system, such as Windows XP or MacOSX, but it's possible to connect with older or more obscure systems.
In the past, early 1990s, you used a modem and the appropriate software, to dial up another modem connected to another computer, and establish a network connection with it. Usually, this computer was linked into the Internet, and so you're online.
With an ADSL modem or router, a similar procedure happens, but a filter splits the telephone line into voice and data (low and high frequencies) and your router negotiates a connection with the ADSL equipment in the telephone exchange.