The Elder Scrolls series comprises of 5 games. The first one is The Elder Scrolls:Arena. The second one is called The Elder Scrolls:Daggerfall. The third one is The Elder Scrolls:Morrowind. The fourth one that you love is called The Elder Scrolls:Oblivion. The last and latest one is called The Elder Scrolls:Skyrim.
Since that is a mouth full we just say there are 5 Elder Scrolls game, Arena, Daggerfall, Morrowind, Oblivion and Skyrim. All those are made from Bethesda Design Studios not to be confused with Bethesda Publishing.
All those games as well as Fallout 3 and 4, do not have multiplayer or any online play.
Now there is Elder Scrolls Online or ESO for short that is made by the owners of Bethesda Design Studios and Publishing and made an MMO (Massive Multi Online) play. There you need an Internet connection all the time because you can only play online. I use to play it, but taken a break for it for now. When I did play I never played it like an MMO with other people I just play it like it was Oblivion or the rest of the Elder Scrolls game, alone. I hope this helps.
Since you are new here, have a fishy stick. http://i0.kym-cdn.com/entries/icons/original/000/002/703/Fishystick.jpg
Here is an explanation so you don't think I am crazy or weird. http://knowyourmeme.com/memes/fishy-stick
Now for some more history form this old fart.
Now to make it more confusing, there is a spin off from Daggerfall that is called Battlespire that does have online multiplayer versus each other in combat. It never took off for it's time. It's an old game so graphics are horrible compared to todays graphics. Also there is Redguard that came out before Morrowind. Both Battlespire and Redguard are not considered part of the The Elder Scroll series though.
Now if you are interested, I believe, Bethesda is still giving out for free Arena and Daggerfall for those who want it. If interested I can see I can find the links for them or someone else can post them.
Davor