What sort of router?
An inexpensive 4 port router would do. They are available with and without wi-fi capability. Get one with the capability. It can be disabled if you don't want to use it, but will allow for future use (great for laptop).
Does this replace my modem?
Again, they are available as modem/router combinations. Since you already have a modem, I recommend a stand alone router. It would be a more flexable setup.
What do i need to do this?
Make sure the second machine has a NIC connection. The 10 meters should not be a problem.