Well, the internet is many, many networks connected together.
A network is two or more computers connected in a way where they can see/speak to each other.
A modem is a product that converts electrical pulses from your(usually) coax cable (think cable tv cable) to Ethernet cable.
Could you connect to the internet with out a modem?
The problem is with RJ45 (Ethernet cables), they are just not good nor built for long distance.
You will almost always need something that converts one signal to another. Which in many cases including this one, is a modem.
Can you get internet without a modem, NO.
If you are trying to hook up your modem to a router, you'll need to make sure that the router is properly configured. Call your ISP and they will walk you through the steps to connect the router to the modem and make sure it is working properly.