So, I am pretty familiar with things when a router is used. To my knowledge, packets travel from Computer A to the router, then the router directs it to Computer B. Problem solved. I get confused when the router is taken out. How can computers within close physical proximity communicate with one another without a router?
Routers route between networks. When two computers are in the same network, they communicate directly together (usually through a switch) without any router being involved.
When two computers are in two different networks, then they need a (or several) router(s).
The subnet mask in the IP configuration is what is used to determine if the target computer is in the same network or not as the sending computer (there's plenty answer on this subject in this site already).
In the simplest scenario, you can connect two hosts directly with each other by Ethernet.
- If the link doesn't come up you'll need a crossover cable. Most NICs support Auto MDI-X nowadays and crossovers aren't required any more.
- The hosts need to be manually configured with static IP addresses (from the same subnet). Depending on the default or configuration, some hosts automatically fall back to Zeroconf aka APIPA addresses (169.254.x.y) which should also work.
- Name resolution likely won't work. Either use the hosts' IP addresses or create a workaround (e.g. using the local hosts files).
Also, you can use any simple Ethernet switch to connect two or more hosts together into a shared network. This removes the crossover problem (if relevant), the rest is the same.
A router is only required when you want to connect that switch to another network.