What network card should I buy?

This question is somewhat dated; most machines only require a single NIC, for which the integrated NIC on most modern motherboards is sufficient. However, machines that perform firewall or routing duty may require additional NICs, and not all integrated NIC chipsets are created equal.

Gigabit ethernet

Gigabit ethernet requires CAT5E or CAT6 cabling to perform well -- I would reccomend paying the slight price difference and getting CAT6, as it has much better electrical characteristics and allows for proper transmission of significantly higher frequencies.

Gigabit ethernet chipsets seem to be of higher overall quality than many of the original fast ethernet chipsets, so in most cases, you can just use whatever chipset may be integrated on your motherboard.

If you need to buy a drop-in NIC, I would get an Intel adapter, since they tend to have the best driver support.

Fast ethernet

Intel and 3com tend to make the best performing fast ethernet chipsets. Both chips will work well under practically any operating system. The drivers are mature and the hardware is well-designed, so CPU usage will be minimized compared to most other chipsets. CPU use is the primary performance concern for fast ethernet.

I generally suggest avoiding Realtek NICs, since they seem to have suboptimal interfaces that make writing high-performance drivers difficult or impossible. They are, however, sufficient for client machines. Keep in mind though that a "PHY" is not the same thing as a "NIC"; the "PHY" merely specifies the electrical interface between the NIC chipset and the network wire. It has no real determinant on the chipset's performance in almost all commercial products. It's not at all uncommon to see a Realtek PHY paired with a 3com or nvidia chipset on a motherboard. This situation should be of no concern.

I highly reccomend getting an Intel or 3com NIC if possible -- I've never had a problem related to one (aside from a dead card, but that happens to all hardware if you use enough parts), and I don't know anyone who has.