If you have a network connector on both. Get a network cable. Plug into network connectors on both. Put both computers in same workgroup. Share the harddrive on old computer. Transfer files as needed.
To connect two computers via direct network cable without a router or switch requires a crossover cable.
I believe store bought crossover cables are usualy red.
I believe there are also computer to computer USB cables but I've never used one.
Before network, we used to use serial or parallel.