Connecting two computers without using a switch or router is actually pretty straightforward. Apart from having two network capable computers you only need some Ethernet crossover cable and ten minutes of your time to get everything up and running.
Assuming both machines have fully functioning network ports, just connect them using the crossover cable. You need to make sure it is actually crossover, and not straight Ethernet. The easiest way to tell, if you have cable lying around, is to check the colors of the connections. If it is a straight through Ethernet cable, the colors will be the same at either end. In a crossover, they will be different.
In a networking environment, the router or switch port will do the crossover internally. As there is no network device in a client to client connection you have to do it manually.
If you want to turn a straight through into a crossover you will need to cut one end, get a new RJ45 connector then change the order in which they are wired. Refer to the diagram for the wiring order of a crossover cable.

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Once you have the cable either bought or made, connect the two machines together and boot them up if you haven’t already.
The vast majority of machines acquire their IP addresses via DHCP, either from a router or domain controller. Because of this we will need to manually assign each machine an IP address in order for it to be able to talk to the other.
So let’s take machine A and set the IP address manually. In Windows this is through Control Panel, Network and Internet, then either Network and Sharing Center, Local Area Connection or right click the network device. Then choose Properties, highlight IPv4 in the little window, then select Properties.