So i have a extra router and i want to know if i connect it via ethernet to my current router which is connected to the modem, will the second router give off internet?
You actually can link routers to each other, but it may require some configuration so they're not fighting for control.
If you're trying to get more ethernet connections, a Switch is the correct piece of hardware, but a Router is also typically a Switch.
You can use the router as a switch. It will give you more ports for internet.
http://www.howtogeek.com/174419/how-to-reuse-your-old-wi-fi-router-as-a-network-switch/