network design

Hello everyone, I am told to make a network and I am bumping into some difficulty in the design. I am told to make a IP plan simply put it's a long list with all the numbers involved in the network, so far I have everything written down I just can't figure out what to do with the router.

this is the physical drawing I made of my network. But I am not sure if this works and why it would work or would not work. I have been told by my mentor to find out what a router does and what and I have concluded is that "a router sends network packages to the appropriate place/device within the network" but that was not a valid answer and he wondered off rambling. So what should I improve in my physical network drawing and why? What does a router do (in simple terms)? and what do I need to document about it(IP address? gateway?)? The database server and the webserver use a peer to peer connection to communicate with each other. Sorry about the stupid / easy question but I just can't figure out what I need to do.