Wowhead's Perculia has put together a great summary that expands on Neth's blog and provides a good deal more detail in the encounters, mechanics and so on that players can expect to meet in the Proving Grounds.
Having spent some time in them today, coupled with Perculia's research, I feel relatively well placed to provide a short run-down on what to expect in the different sections and difficulties. First up, where are they? The proving grounds are located in the same spot as the Arena of Annihilation, the White Tiger Temple in Kun-Lai Summit. If you get there and can't find the NPC, and instead see a white tiger, you are phased and need to complete the opening of the Vale quests. Fortunately, it's only three quests. Once you're there, speak to Rotun and get started.
The tanking challenge puts the player in a party with Sikari the Mistweaver, who will take care of both DPS and healing for you, ticking for 70,000 every so often. She's neither a great DPS nor a great healer, refusing to pop cooldowns to keep you alive, and seemingly unwilling to heal herself on occasion, but nonetheless she's all you have, so keep things away from her.
The tank challenges center around keeping various mobs at range of Sikari, retaunting from her when knocked back, avoiding AoE and interrupting casters. There are various adds of varying levels of strength and annoyance, the worst of which are the banana adds, who fling out a damaging AoE and seem to appear mid-wave. It's certainly challenging, and fun, and a good way to learn to tank and to master new tanking classes.
As a healer you will be paired with a tank and 3 DPS characters. These appear in whatever party frames you're using, and are healable with clique, vuhdo and healbot. I didn't try any other addons, because I don't have any! But nonetheless, they're easily healed.
As a healer you're going to be dealing with a DoT to dispel that does 50% of health if allowed to expire, party-wide AoE damage, interrupts (if you can, your party will interrupt too), DoTs that require a player to be healed to full to dispel, additional tank damage from an add enrage, and a hunter who repeatedly stands in bad! It's all highly personal, but the healer section seemed a little undertuned compared to the tank and DPS section, but that could be just because I'm much more used to playing healers.
The DPS section has you as a lone wolf facing off various foes. In order to introduce some interest, you have a timer for each wave of adds, and if the timer expires before the adds are dead, you will explode. As you can likely imagine, that's something to be avoided!
There are several types of add for DPS too, ones with shields that mean they have to be attacked from one side or the other, AoE adds, healers to be interrupted, amber-weavers who fling balls of amber at you that will basically put a freezing trap on you if allowed to hit you. These should be kited into other adds. The DPS one seemed fun, probably somewhere between the healer and the tank in difficulty.
Overall, these are great fun. The endless mode, which takes you to 30 waves, awards a title, The Proven [Role], and all the difficulties award achievements. Endless also features a score, which is tracked as you go and planned to be visible by others. Do check out Perculia's post on Wowhead for more information.