The FIRST game I played across a network was Doom (DWANGO over modem!), using my hayes compatible 14.4 KHz modem. The first game I played "online" was Duke Nukem 3D, using a third party application that allowed online (albiet, terribly laggy) games. The first game I truly played online without odd caveats was Quake, fallowed quickly by the popular mod Team Fortress (and later I tended to prefer TF Mega, which altered the rules somewhat).
My first MMORPG was Arctic, a MUD. My first graphical MMO was Anarchy Online. The first game I played on high speed internet was Counter Strike. The first game I played on XBL was COD 4.