You can use Linux TCP/IP without any networking hardware at all---configuring ``loopback'' mode allows you to talk to yourself. This is necessary for some applications and games which use the ``loopback'' network device.
However, if you want to use Linux with an Ethernet TCP/IP network, you need one of the following Ethernet cards: 3com 3c503, 3c503/16; Novell NE1000, NE2000; Western Digital WD8003, WD8013; Hewlett Packard HP27245, HP27247, HP27250.
The following clones are reported to work: WD-80x3 clones: LANNET LEC-45; NE2000 clones: Alta Combo, Artisoft LANtastic AE-2, Asante Etherpak 2001/2003, D-Link Ethernet II, LTC E-NET/16 P/N 8300-200-002, Network Solutions HE-203, SVEC 4 Dimension Ethernet, 4-Dimension FD0490 EtherBoard 16, and D-Link DE-600, SMC Elite 16.
See the Linux Ethernet HOWTO for a more complete discussion of Linux Ethernet hardware compatibility.
Linux also supports SLIP, which allows you to use a modem to access the Internet over the phone line. In this case, you'll need a modem compatible with your SLIP server---most servers require a 14.4bps V.32bis modem.