How is the input connection made to a punch down block?
Ethernet cabling isn't simply jointed together the way power or phone cables are. Each run from the various rooms should be punched down to the back of an ethernet patch panel:



It's unclear from your question what kind of patch panel you have or how you've punched down the cables. An ethernet patch panel will expose a standard RJ45 connector for every line. Then you need to connect each run to a different port on an ethernet switch: 



In your case you will need to buy a switch with at least as many ethernet jacks as you have in the house.

A switch is a device that directs internet traffic to and from the various ports based on their source and destination. It is not just a dumb splitter. You then plug the router into one of the jacks (doesn't matter which one). Note that a modem and a router have different purposes, although occasionally they are bundled into the same device. You must make sure you have a router either built-in to the modem or sitting just after the modem.