Well if LAN counts I think it was Starcraft, otherwise Diablo 2 was the first game I play over the Internet. Although on crappy dial-up I use the word "play" loosely.

Descent I direct connect via 14.4 dial-up modem

Chess, ChessMaster I think, and Starseige

In 1998, my first online game was Warcraft II hosted on AOL's Engage Server. In the AOL community associated with the Engage Server, there were message forums dedicated to Warcraft II and the moderators there supported and encouraged the formation of online Warcraft II "clans." This was also my first encounter with online gaming clans. Existing clans on these Engage forums were unofficially classified by players into three types: The Horde, The Humans and Neutral factions. Major clan names I can recall off the top of my head were: Nation of Calderan, Splintered Orc Clan (SOC), Death Raiders Clan, United Dark Forces (UDF), Dark Covenants (DC), Keepers of Edanvant (KoE), Descendants of Kilrogg (DoK), Banners of the Mark (BoM), and my own clan, Death Falcon Company (DFC).

Total Annihilation in 1997 baby

Quake 2, with a 56kk modem lol...

First Online Game; Ultima Online played over 56K modem. It was painfully slow.

Current Online Game: DDO (because its free)

"Starsiege: Tribes.

It was such a fun game. I wish they'd make a updated version (and I don't mean Tribes 2)"

http://en.wikipedia.org/wiki/PlayTribes

Diablo