How can a desktop computer connect to the internet without a modem and router?

You usually have to have a modem and an internet provider, you don't need a router.
You need to connect your modem to the cable or optical fiber entering your house and then plug your computer directly into the modem with an Ethernet cable or optical cable unless you can tap into somebody else's WiFi if your computer has a WiFI card or integrated WiFI.

The modem can be a coaxial type or an optical network terminal if you have an optical input to you house.

A router just gives you a wireless a WiFi set up for other computers TVs, etc, in your home..