The intent of Terramorphous is to be a raid boss, something you bring 4 players along to fight at once. It's possible to beat him with a coordinated team, good equipment, proper skills and tactics (or a reasonable combination of the above), but there's a super cheap slightly helpful way to beat him even solo, too.
The Cheap Way
With The Bee shield and the Conference Call shotgun, Terra is trivial. The Bee adds 40k-50k extra damage per bullet, meaning any good shotgun can do over 500k damage to Terra in one shot. The Conference Call shotgun is optional, but shotguns are best for this method, and the Conference Call is ideal.
The Bee has been significantly nerfed in a recent patch (actually all Amp Shields were) so the amp damage no longer applies to each bullet from multiple bullet weapons (shotguns); instead the damage is split across all projectiles. This means shotguns went from insanely broken with The Bee to being only minorly boosted. Instead try a high Rate of Fire weapon that shoots single bullets per shot (or that fires pellets so close all of them will hit).
You can get The Bee from Hunter Hellquist.
Rapid Fire guns like SMGs work very well. If you can boost shield recharge rate at all, do it; you'll want your shield charged as often as possible, the Bee only kicks in at max shield.
Killing Terra is much harder than it used to be with The Bee but it's still a relatively quick way to do it. The more the merrier though; a team is now almost required so see the tips below rather than hoping on soloing him.
Tips for doing it the right way
Like the mission description says, You Will Die. Even if you take the cheap route, this boss can kill you with a flick of his wrist (tentacle).
Be level 50/max level
This should go without saying, but it's vital. Terra will always spawn at level 50-52. You deal decreased damage to targets higher level than you, so even if you have The Bee and amazing weapons, your damage will be drastically reduced if you're level 45 and Terra is level 52. Save this fight for after your second playthrough to save yourself a lot of grief.
Communicate
Everyone should use headsets if possible. You don't have time to type (don't try, you'll get killed and piss everyone off), use headsets or just play in silence. Coordination is key for doing things live reviving allies
Respec your skills
Lots of skills are useless in this fight. Kill skills aren't great (but you will have tentacles to kill), anything that requires multiple kills isn't great. Elemental damage isn't helpful. Abilities that benefit your team over yourself are often helpful, like the ability that turns Phaselock into an instant revive. Note that no part of Terra can be phaselocked, so keep Phaselock for Revives. Use Decepti0n to trick Terra while reviving allies.
Hug the wall
There's a cliff face on one side of the arena; all players should gun this wall unless running from an attack or grabbing ammo/health (tentacles drop a lot). Spread out a bit so he can't kill you all at the same time. Standing high on the outcropping near the wall will avoid Terra's direct attacks. There's also a little niche where one player can hide from some tentacle attacks. If you don't hug the cliff, Terra's knockback can often knock you off the edge of the arena for an instant kill.
Manage your respawning
Like Crawmerax before him, Terra has a slow elevator leading up to him. After being used it takes a minute or two (seriously, not "a few seconds" or "a little bit") to shoot all the way up. If two players die, make sure you both get on the elevator at once or one of you will be waiting a long time to get back in.
Get back in the fight as soon as possible though, as soon as no single player is alive (or in Fight For Your Life) in Terra's arena, his health resets to max. Be quick but be reasonable; if another player is about to die, wait a couple seconds for them to spawn and use the elevator together. Buy ammo if absolutely necessary, but remember his tentacles drop lots of ammo anyway.
Elements
Terramorphous is occasionally elementally charged and will resist the element he shows up as; I've seen him as Shock and Fire, not Corrosive yet but I'm not sure. It's best to not bother with elemental guns, he doesn't appear to take Damage Over Time damage and doesn't seem especially weak to anything. Sirens should spec away from elemental damage.
Terramorphous can occasionally be slagged but he usually resists it. When he's slagged it lasts for about one second instead of 8, so don't bother unless your best gun is already slag.
Guns
Obviously use your best guns. Don't use elemental guns hoping they'll do DOT, but he does seem to take elemental damage normally. There's no particularly fancy strategy here, put a lot of very big bullets into him. Bring different kinds of guns or you'll run out of ammo. Bring at least one long range gun for the tentacles.
Being a Thresher Terra moves a lot and has many relatively small crit spots, so sniping can be trickier, but due to his bulk his crit spots aren't too small.
Wound tentacles, don't kill them unless you need to
The tentacles he spawns can be annoying, but they're great for Second Winds. Generally it's best to just wound them or ignore them, depending on type, and use them for a Second Wind if someone is knocked out. Hurting tentacles hurts Terra, too, but unless you're sniping you're likely to miss the tentacle a lot more than Terra itself.
Note tentacles drop lots of health and ammo though, so kill some just to keep everyone full of ammo.
Loot
Terra drops a lot of whites and greens, a few blues, a couple purples and always drops one orange class mod (it's not as good as you might hope), and apparently has a chance to drop other orange weapons.
Unlike Warrior, Terra drops all his loot in a tight circle around where he dies. There's pretty much zero chance his loot will fall off the map or be stuck in his body, making the looting much less frustrating than Warrior or BNK3R. You can also see all of the items at once, so there's less risk of ninja-looting ("oranges? There's no orange guns over here....")
Comparison to Crawmerax
I'd say Terra is easier than Crawmerax, even ignoring The Bee trick, and certainly less frustrating to fight. His tentacles will rarely kill you unless you fail to evade, as opposed to Craw Maggots which would zip in, kill you, and flee out of range. Getting second winds is easier, and you don't have to worry as much about elemental matches with the minions. The lack of specific weakpoints you have to kill, one by one means it's much easier to hurt Terra; just shoot in his general direction. He soaks up a lot more damage, but it's much easier to hit him.
If player respawns are spread out, getting back into the fight isn't quite as long or annoying as the infamous Crawmerax elevator. And he doesn't drop 50 %$^&ing equalizers every time he does. Generally he's still extremely hard like Crawmerax, but not frustrating. The only annoying "Crawmerax" thing Terra retains is the instant-kill attacks and knocking players off the map (instant kill without a chance to revive!).
Best Answer
Seraphs used to drop Seraph Crystals in Playthrough 2 when that was the last playthrough available; with the addition of the Ultimate Vault Hunter pack, there is a new last playthrough (#3), so they only drop the crystals in that playthrough.