A server is just another computer. Therefore, you don't need one in order to have a network between your two computers. Instead, you have a choice:
You can purchase a switch and connect your both machines to the switch.
The cheapest switches cost less than ten dollars, and allow to plug five or eight devices. Note that you also have to have two Ethernet cables. Ensure first that both of your computers have a network adapter.
Or you can buy a crossover cable to connect the computers directly without a switch.
While this solution is cheaper, it is also the least scalable. If, in the future, you would want to connect a third device, you wouldn't be able to do that if every device has only one network adapter.
If you have a router, there is possibly a way to either connect two devices through Ethernet to it, or to use Wi-Fi. In this case, you don't need to purchase anything extra.
Once you have your local network set up, you can start exploring the capabilities of distributed computing.