A little history about resilience
Way back in the day, in the vanilla WoW, there was no resilience. PVP and PVE gear was one and the same, outside of particular items you could by from reputation and climbing the PVP ladder. Some of the gear may have a little more stamina, but for the most part, PVP gear used the same stats as PVE gear.
Then came the combat rating system, Arena, and The Burning Crusade. Resilience was introduced as a combat rating that would reduce incoming damage from other players. The exact methods of what damage and how has changed over the course of two expansions, but the general theme has always stayed the same. Resilience reduces damage caused by other players.
The original resilience stat didn't cost much from your itemization budget, though. That means that just because a piece of gear had some resilience on it, that gear didn't automatically become a gigantic liability. Hell, some people even did the arenas expressly to get PVP gear, which was perceived as worthwhile and easy to get.
This situation could not stand, of course. Both to alleviate complaints that PVP gear was too easy to get for too much value in PVE and to create a larger separation between the two playstyles, the itemization cost of resilience was radically increased.
What resilience does for you
Resilience has gone through some Light-awful gyrations since its inception. It once reduced crit damage, crit likelihood, and various sources of damage at once time or another. Now, though, it does one thing in very simple terms: It reduces damage from other players.
This damage reduction is reported on your character sheet under Defense. Resilience on your gear is actually a rating, so that's the best place to look for your final resilience effect.
Do you need resilience?
So now that you know more about resilience, we have to discuss whether or not you need it. The very short answer here is a simple yes. By the same token that resilience is well-nigh useless in a dungeon, it becomes mandatory in PVP.
We all have more stamina in the current expansion than we once carried, but we also do a whole heck of a lot more damage. Even a tank with 200k hit points is a barely over 10 seconds of work for a mage who's only half trying. You simply must have resilience to survive once anyone starts hitting you.
I've seen some arguments about people eschewing resilience gear in the Battlegrounds, and I can see some arguments. If you're tanking an Alterac Valley boss, for example, you'll want some tank gear. But really, isn't that encounter more competitive PVE than actual PVP?
The rule is simple. If you're getting hit by players, you need resilience. Without it, you're the player-guided version of a road bump.
Of course, the problem with any kind of rule is that it no sooner leaves your mouth than exceptions start happening. Still, the exceptions to needing-resilience all center around the fact that they're not really getting hit by other players.
A rogue could argue he doesn't need resilience, especially if he's the sort of rogue who hangs out at the edges of a fight waiting to snipe a healer. If he's sufficiently skilled at his craft, he might be about to pop out of stealth, grab a kill, and disappear again.
Some Arena teams might strategically eschew PVP gear in order to increase outgoing damage. I do 2v2 with my wife sometimes as a retribution paladin and a healer. I sometimes feel like I'm wasting itemization by using resilience, because every team trains on the healer, every time. I've tried everything short of draping pork chops around my neck to get the other team's attention, but until the healer's dead, I just don't get hit.
Still, you'll see the theme. If you are in a situation where you are not getting damaged by other players, then you can consider skipping it.
How to get it
Getting resilience gear is thankfully pretty easy. It comes nearly automatically by doing PVP content. If you do Battlegrounds, you'll quickly pick up all the honor points you need to go buy the gear. You'll need to do Rated Battlegrounds or Arena to get enough conquest points for the high-tier stuff, but the honor gear will get you started.
Use as intended
As a final note about PVP gear and resilience, please refer back to the idea that all of this gear is intended for player-versus-player content. While you can use the stuff to circumvent the item level requirements and supercharge yourself into a Troll instance, please don't. It just puts PVP into a bad light.
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 learning how to tank, getting up to speed for heroics and even how to win Tol Barad.