Probably this is not a problem but I do not know what to do. I would like to connect two computers in a network and to enable them both to access Internet. I do not have a router and I am not in the mood to buy it. A friend told me that I need two LAN cards then. Is this a true and how am I going to configure them?
You are right and this is not a problem. Two LAN cards can function in one computer and it is the only way to connect two computers to Internet at the same time without the router or server. You need crossover cable to connect two computers and use another NIC to connect modem. Configure Internet connection sharing on computer with two LAN cards and you are ready to go.