You can connect it through a traditional network cable. The computer needs a network card for that.
It is still possible to get dial-up service over a telephone line. You can search for a local ISP (internet Service Provider) that still supports dial-up access. Net-Zero is one such company. This requires a modem port on your computer. (don't confuse with an ethernet port, the connectors look similar, but the ethernet connector is wider than the phone line connector).
On a windows system you can check for a Modem or Network adapter with the Windows Device Manager.
If you have a Network adapter, usually some ethernet controller, you can get ADSL or a cable modem. If you don't have a network adapter (with an ethernet port). You could buy one.
ADSL service uses your phone line, it splits the signal so you can use your phone and computer at the same time. Contact your local ISP.
A cable modem, is provided by some TV cable services.
With both ADSL and a cable modem service connect to your ethernet port.
If you have USB ports you can buy a WiFi adapter and plug into a USB port on your computer. (You can also buy a WiFi card and plug into your computer chassis. This will require that you have access to a wireless network. You can buy a ADSL modem with wireless, or add a wireless access point to your ADSL or cable modem.
3G service is available, but expensive.
Summary, find a local ISP, they can help you.
Dial-up is very slow, and not very useful, downloading pictures, updates, new software is very, very slow.
Think about adding wireless to your computer via a card or USB adapter. It means no ugly cables to manage, can handle multiple computers.
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