You'll first encounter Terramorphous the Invincible at the end of your 1st playthrough when the mission "You. Will. Die. (Seriously)" will become available from Tannis after you've defeated the last boss and the credits have finished. He will be located at Terramorphous Peak in the Thousand Cuts area and he's not an easy boss.

To have any chance of beating him, you'll need to be at least level 50 and will need a very good Badass Rank. As well as that, you may need 3 other players on your team who are matched the same in level/stats. It's also best to have one of each character in some situations, because of certain Skills working best off each other.

One thing you'll have to make sure you don't do is let everyone die. If everyone in the arena fighting Terramorphous dies, his health will be reset back to 100%. Plus every time you want to fight him and enter his lair, it'll cost you 4 Eridium. Other than that, good luck.