If you've ever played a caster class in WoW, you've learned to hate "spell pushback". A melee character can be taking massive amounts of damage, but they can always fight back (as long as they're in melee range). Hunters have an odd hybrid of mechanics in that their ranged attacks actually cut off if their foe is within melee range, but they have a pet and various snares/traps to try and keep that from happening. Casters can always cast their spells to deal damage... but taking even a single point of damage per second can slow the fastest spells to a crawl.
This matters less in groups; typically most damage can be either tanked or avoided (e.g. "get out of the fire" fight of your choice), and only a few fights actually spew out unavoidable damage on a regular enough basis to affect your casting (the fire pulses on Vael and Firemaw in BWL come to mind). PVP is a different beast entirely, where the prevalence of fast attack speeds and interrupts are going to strongly favor instant cast spells (which cannot be delayed or interrupted) when you're under attack, with more time to cast real spells while the enemy attacks your allies. So the real question is what happens when you try to solo, where you've got no one else to take that damage for you.
Since my most recent round of PTR premade testing, I've had at least some experience trying out all of the DPS caster specs in the game. (I'll admit that my knowledge is far from perfect; indeed, I'm learning on my Shadow Priest that I simply don't like the spell Mind Flay all that much because of the channel bar.) One thing that has impressed me is the wide variety of different methods designed into the game for mitigating a mechanic that is only there to begin with because the developers put it there. Here are the major methods:
Outright Resistance Via Talents
Many classes have a talent somewhere in their various trees that confers partial - usually 70% - resistance to interruption due to damage. This number is high enough to let you get off that last spell needed to finish your foe, but deliberately low enough that you won't be able to cast if you're tanking multiple mobs. There are a few exceptions that actually go to 100% immunity with talents, notably a Mage's Arcane Missiles (the spell is nigh useless solo without the talent, due to its high mana cost and periodic damage) and a druid's Entangling roots (my next point). Note that these options are not always available immediately, which means that sometimes you have to level into your mid-20's before getting the full advantage of talents. There are also some temporary abilities that offer full pushback immunity, e.g. in response to a critical hit.
Don't get hit
Obviously, you won't lose casting time to damage if you don't take damage. The Warlock accomplishes this task by sending in a pet to tank. All mages have access to Frost Nova for a brief root, and Frost spec mages get to freeze foes more often (and do more damage to them while frozen). Roots are a sufficiently important spell for Balance druids that they're allowed to gain 100% un-interruptibility (and the spell is slated to become usable indoors in Wrath). Kiting, the fine art of snaring the foe, running far enough away to cast before it gets to you, and repeating, also falls into this category, though you're going to want instant cast spells at your disposal if you're trying to cast them on the run.
Cast faster, noob!
Casting faster is more commonly thought of as a DPS buff (more spells in the same amount of time), but it also means less total time before the spell goes off. Talents, spell haste rating, and some procs/abilities/trinkets help in this department. Faster casting can also mean picking spells with shorter cast times. A mage's 1.5s Scorch cast becomes nigh un-interruptible when combined with 70% immunity from fire talents (which are even available to non-fire mages, such as my unorthodox Water Elemental/Master of Elements solo build).
DOT and forget
If you can't cast faster, the other approach is to take your time. Many damage over time spells can be cast instantly, at the cost of taking time to deal their full amount of damage. There are also some spells that appear to be designed as openers, notably a Mage's Pyroblast or a Priest's Holy Fire, that have lengthy cast times but apply a DOT effect in addition to an initial hit. If you're lucky, you can do almost your full amount of damage while minimizing time spent "casting".
Take it on the chin
Well, if you must cast spells that are not pushback resistant and you cannot keep your foes out of melee range, your only option is to try and eat the damage somehow. The iconic form of damage absorption is a priest's Power Word: Shield, though Frost mages get a self-only clone of the ability.
This last part is where I was having problems on my shadow priest, as I was trying too hard to use Mind Flay (which people say is great), and channeled spells really suffer if you don't have some way not to lose casting time to damage since you're paying the full mana cost up front. I simply hadn't needed to keep PW:S up at low levels (my wand DPS is good enough that I'd rather stop casting and benefit fully from Spirit Tap), which was fine when I was just losing casting time but less fine when I started losing mana to mind flay. Sure enough, I altered my strategy a bit to avoid relying on mind flay in situations where I was taking damage with PW:S down, and suddenly I was having a lot less trouble playing the priest.
Like I said at the top, I'm stuck by the variety of all the different caster specs/strategies. It can't have been easy to design each of the classes so their abilities diverged like this, but Blizzard managed it somehow.