PS2 online required a Network Adapter (fat PS2) or built-in Ethernet (slim), a broadband connection, and—critically—most game servers are now dead. The following titles had online modes via Sony Online Entertainment (SOE), GameSpy, or EA Nations:
Official servers for all above are offline. SOCOM I/II and Twisted Metal: Black Online have active fan-run replacements. For everything else, only LAN/local play or single-player remains.