If the interface of your NIC is found in dmesg, skip this section.
If your computer has a NIC that may not be supported, you still have a few options. First, from the lspci or lsusb listing, write down the name of the vendor and the name of the chipset. Next, on a Internet connected computer, use a search engine to look for keywords including the name of the distribution (such as Debian), the name of the chipset (such as rtl8168) and optionally the name of the vendor (such as Realtek).
If the search results come back positive, you can proceed to download the proper packages and install them. Otherwise, you may need to consider permanently adding a NIC that is supported. In either case, you need to install a supported NIC to the computer to make this process easier.
If an additional NIC only needs to be added temporarily to install the necessary drivers for the permenant NIC, you can consider using a USB NIC. This way, there is no need to open the case of the computer. A 100Mbps USB Ethernet card is inexpensive, and it should suffice to install additional components.
On the other hand, if you need to install another NIC permanently, consider installing a PCI or PCIe NIC. This is because a production system may need more network speed than that supported by most USB NICs.
To install additional packages in a Debian system, you may need to edit /etc/apt/sources.list to include contrib and non-free after the word main in the main repository entry. After this step, you need to update the list of available packages by running aptitude update.
After activating additional repositories, you can then install the necessary packages. If you don’t need to keep the extra NIC, remove it before starting the system again. After rebooting, go back to the first section to check the system log and see if your network card is recognized by the system.