Sluissa is somewhat correct, though his unflattering depiction of armor is universally wrong. Also, surrounding a battleship with countermeasures vessels does NOT take away their agility---it's not a very agile ship to start with.
So, US battleship armor: More than a foot of armor on the belt, a lightweight deck with a "splinter deck" beneath it, and lots and lots of antiaircraft guns.
Now, the belt armor will stop a LOT of missile damage. However, battleships can't take that fire without issues and they can't take hits like that indefinitely without suffering lots of damage, which is why the carrier is so much better at taking on aircraft.
The deck armor will stop non-armor-piercing-bombs perfectly well, and will prevent a lot of damage from AP munitions as well. Not that dive-bomber-launched armor-piercing bombs are a thing, anymore, now it's all fancy GPS-guided missiles/bombs/etc. The deck is going to be something of a weak point.
As for everyone's points about the uselessness of naval bombardment---that's CUTE. Alright? CUTE. Battleship guns are more accurate than you give them credit for---capable of sinking a moving target at 20 miles or MORE, firing only based on radar. When the target isn't moving, and the ship isn't moving? They're DEADLY at ground support, capable of staying on target for MONTHS compared to the hours or minutes of a regular plane. They can rain down shock-and-awe like nothing else in the world's arsenals, and they can do it much, MUCH faster than aircraft, which have to be sortied, flown to the target, and THEN they hit the target. A battleship lines up the guns and fires, and within three minutes any target within their range is getting hit with massive explosives falling from the sky. There is a REASON that the US reactivated the Iowas for Vietnam, Korea, and the Gulf War, and it's not because they were useless.
As for carriers, that's highly inaccurate. A carrier can be EASILY crippled with a single lucky hit. If the deck (Specifically the catapults) receives damage, the carrier's done for. With a battleship? You can take antiship missiles to the side, have an entire turret get knocked out by guided missiles, and keep on firing.
It's what they DO---it's much, MUCH harder to sink a battleship than it is to sink a carrier, provided you can reach both equally well. That's the carrier's strength, and the reason that a battleship is better at shore-support (Within 24-26 miles, anyway)----the carrier hangs out further away and can attack, but takes more time to attack. Yes, a carrier strike can hit a lot harder than a single battleship salvo. But in the hour and fifteen minutes to launch one attack and fly to the target, the battleship has gotten (85 minutes x 2 rounds per minute x 9 guns = 170 * 9 = 1530 shells) enough salvos off to fire every shell in its ammunition bunkers, and every shell in the bunkers of several comparable battleships.
Which is a LOT more destructive firepower than a single carrier sortie, even if we bow to reality and say that the ship has PROBABLY only fired off half the shells in its bunkers because the target is nothing more than a set of large craters in the landscape.