Hmm. I started with Doom II for my first Multiplayer game, and it was a blast but for the best?
I would say its a few games.
Half-Life (original) deathmatch.I played that for hours upon hours before TFC came out, and it was a blast, it had some of the greatest maps, and weapon balance.
Duke Nukem 3D and Blood. These games had some of the most unique weapons that created some of the most entertaining matches I had ever had.
Counter-Strike-Started playing in Beta 1 and loved it right away, played the game for almost 4 years then it got so cheater "filled" I left. But a truly superb game.
But one of the best, that I think opened up the idea of larger communities and contributing to it was the original Quake, and I guess Quake II could be put in there as well (though some of its maps were way too big). But I remember Quake's carnage, and just intense adrenaline filled action. You had to be quick, and know the maps. Ahhh I miss the days of rocket jumping. But also the mods were amazing for these two games, Team Fortress on Quake really changed everything, and was so addicting. It was my most fond memory online.
As for another mention I will say Unreal Tournament (original) and 2004 to a lesser extent. But the original UT I remember just being overwhelmed by all the customization you could do, it became a huge LAN game my friends and I played. And of course online as well.