Don't be a mana sponge
You can probably see the issue right away. If a healer's mana is limited, then you need to conserve it as best you can. Sure, healers are tasked with having as much mana as possible, using their heals efficiently, and basically doing what they can. But there's a lot of synergy between a tank and a healer, so you as a tank should absolutely do your part in this equation.
A tank who has gemmed and enchanted for stamina in lieu of mitigation is a big, gigantic mana sponge. Certainly, you can take big hits from the boss; after all, you've got all these juicy hit points lying around. But the hits you take from a boss will be bigger than those on the tank who has mitigation, and you're therefore actually much, much more difficult to heal. You will soak up the healer's mana, and you will run his mana bar dry.
If you want or need just a little more proof that a bottomless well of hit points isn't necessarily great, consider General Umbriss in heroic Grim Batol. He has an ability called Bleeding Wound that cannot be removed until the tank is healed to 90%. You definitely make this fight easier if you've focused on mitigation instead of stamina.
What is mitigation?
There are four types of mitigation you can pick up as a tank: armor, dodge, parry, and block. Armor tends to be the "best" kind of mitigation because it reduces the amount of damage you take from all physical attacks.
Dodge and parry are about the same critter nowadays. The amount of mitigation percent you get from each stat is the same, though it should be noted that bears can't parry. (They have no weapons, you see, and we don't yet have the talent Iron Paw.) When you dodge or parry, you take no damage from an attack.
Block is a little more complicated; warriors and paladins each have the ability to block incoming attacks with their shields. Whenever you block an attack, you mitigate the same percentage of damage. Warriors can, additionally, get a critical block to mitigate a little more.
So, armor reduces damage off the top. Dodge and parry let you completely avoid the incoming hit. Block reduces the damage by a percent.
Devil in the spells
Unfortunately, you can't block, dodge, or parry spells. Each tank class has different methods for dealing with magic damage, whether it's a talent like Perseverance or a skill like Anti-Magic Shell.
Know your class's method for dealing with magic damage. Love it and embrace it. Buy it flowers, and let it know occasionally how meaningful it is to you. That you think it's special, and awesome.
But you may also need to keep a stamina set on the side for fights where the damage is just so huge, ridiculous, and all-magic that your normal tanking set can't keep up. After all, if you have 130k hit points and you get clocked for a 140k spell, all the block, dodge, and parry in the world won't help you.
Ask your healer
I have a macro I use often when I'm tanking PuGs. It tells the group, "Let me know if you're having challenges healing me; I have different pieces of gear and can switch them out according to the situation."
As you get further in your adventures as a tank, you'll find yourself carrying "threat gear," "stamina gear," and "avoidance gear." In PuGs, I tend to go straight to threat gear, because I've not met a PuG yet that likes to wait for a tank to slowly get aggro.
If the healer tells me I feel squishy, I start swapping into my "avoidance gear." And I keep doing it until we've reached a happy medium.
Healers tend to have a great instinct for "squishy" versus "rock-hard abs." They're the ones staring at your health bar, after all. I think of my health bar as belonging to the healer; I'm protecting it, but they're the one tending to it. So ask their opinion and solicit their input.
Visit the WoW Rookie Guide for links to everything you need to get started as a new player, from how to control your character and camera angles when you're just starting out, to pulling together enough cash for mid-level expenses such as mounts, to dungeoneering and travel tips for lowbies.