With the new MTG set only a few of weeks away here is a breakdown of the mechanics we can expect...
BOLSTER
Bolster is the signature ability of the green-white Dromoka clan. It hasn't changed since its appearance in Fate Reforged.
To bolster, first find the creature you control with the least toughness. If there's a tie, choose one of those creatures. Then put a number of +1/+1 counters on that creature equal to the bolster number. Bolster doesn't target any creature. You don't determine which creature is getting the counters until you're taking the bolster action.
So suppose you control a 2/2 creature and a 1/1 creature and you cast Sandcrafter Mage. When Sandcrafter Mage enters the battlefield, its ability triggers. In response to that ability, your opponent destroys your 1/1 creature. When Sandcrafter Mage's triggered ability resolves, you control two 2/2 creatures. Since both of these creatures are tied for the least toughness, you choose one of them. It gets a +1/+1 counter, and you have a shiny new 3/3 creature.
REBOUND
Rebound is the signature ability of the white-blue Ojutai clan. Rebound is making its return (you all saw this coming, right?) after originally appearing in the Rise of the Eldrazi set. How it works hasn't changed.
Rebound lets you cast an instant or sorcery spell a second time for free. If you cast a spell with rebound from your hand, you exile it instead of putting it into your graveyard. Then, at the beginning of your next upkeep, you can cast the spell again without paying its mana cost. After casting it from exile, it's put into your graveyard. If you manage to have several spells rebound during the same upkeep, you can cast those spells in any order.
If you have any creatures from the last two sets with prowess, casting a spell with rebound will give you two bonuses, once for each time you cast the spell. (Prowess doesn't appear in Dragons of Tarkir.)
Rebound works only if the spell resolves. If it's countered for any reason, either because your opponent casts a spell like Negate or because all of the spell's targets became illegal, it won't rebound. It will just be put into your graveyard with no effect. If you can't cast the spell from exile, perhaps because it requires a target and none are available at that time, then the spell just stays in exile.
EXPLOIT
Exploit is the signature ability of the blue-black Silumgar clan. When a creature with exploit enters the battlefield, you may sacrifice a creature you control.
But you're not just sacrificing your loyal minions for fun. Each creature with exploit has another ability that gives you a benefit when it "exploits a creature." This means when you sacrifice a creature because of its exploit ability. That ability doesn't trigger if you sacrifice a creature for any other reason, including the exploit ability of a different creature.
You can sacrifice any creature you control when the exploit ability resolves, including the creature with exploit itself. You don't have to sacrifice a creature if you don't want to. If you do, you choose which one as the exploit ability resolves. To get the most out of your minions, look for creatures with abilities that give you an added benefit when they die.
DASH
Dash is the signature mechanic of the black-red Kolaghan clan. Dash allows your creatures to strike and disappear before your opponent knows what's happened. It hasn't changed since its appearance in Fate Reforged.
Dash is an alternative cost found on creature spells. As you cast a spell with dash, you can pay its dash cost instead of paying its mana cost. If you do, the creature will have haste, so it can attack that turn. At the beginning of the next end step, you'll return the creature from the battlefield to its owner's hand.
If you choose to pay the dash cost, you're still casting the spell, so it goes on the stack and can be countered. Dash doesn't change when you can cast creature spells, usually only during your main phase. If you cast a creature spell using its dash cost, it will return to your hand only if it's still on the battlefield at the beginning of the next end step. If it leaves the battlefield before that point, it'll just stay wherever it is.
Some dash costs are lower than the mana costs of the creature cards they're on. So, if you have four mana available you can cast Sprinting Warbrute a bit earlier than you otherwise could and get an extra attack in. On the next turn, you can cast it for its mana cost and it'll stay on the battlefield like normal. Other cards have dash costs that are the same or higher than their mana costs. Attacking with such a creature right away can be powerful, so you may want to cast such a creature using it dash cost even if you could pay its mana cost.
FORMIDABLE
Formidable is the signature mechanic of the red-green Atarka clan. Formidable is an ability word, so every formidable ability is different, but they all care in some way about controlling creatures with total power 8 or greater. Read each one carefully to see exactly what it does.
When calculating the total power of creatures you control, use their actual powers, even if they're less than 0. For example, say you control creatures that are 0/3, 4/4, and 8/8. The total power of creatures you control is 12. But if your opponent cast a spell that gave the 4/4 creature -6/-0, you'd have an 0/3, a -2/4 and an 8/8. Their total power is 6.
Stampeding Elk Herd's ability has an "intervening 'if'" clause, which is basically a condition of a triggered ability sandwiched in the middle. Abilities like this check the condition two times: once when the ability would trigger. If you don't control creatures with total power 8 or greater at that time, the ability doesn't even trigger. If the ability does trigger, it checks whether you still control creatures with power 8 or greater to see if it resolves. If you don't, the ability does nothing.
Other formidable abilities are activated abilities. You must control creatures with total power 8 or greater in order to activate them. However, once you activate such an ability, it doesn't matter what happens to your creatures. Even if their total power dips below 8, the activated ability will be unaffected.
MEGAMORPH
Think of everything you loved about the morph ability from Khans of Tarkir. Megamorph is all of that, but bigger!
If you understand how morph works, megamorph is almost exactly the same. The only difference is that if you turn a face-down creature with megamorph face up by paying its megamorph cost, not only will it suddenly have its normal characteristics, but you'll put a +1/+1 counter on it as well! Note that if you turn a face-down creature face up some other way (for example, say you manifest a card with megamorph and turn it face up by paying its mana cost), it doesn’t get the counter.
Turning a creature face-up doesn't use the stack and it can't be responded to. So if I controlled a face-down Aerie Bowmasters and turned it face up, it would be a 4/5 (3/4 with the +1/+1 counter) before any player could do anything.
The morph overlay returns in Dragons of Tarkir booster packs to place on face-down creatures with megamorph. It's not necessary to distinguish between creatures you cast face down (although distinguishing between them and manifested cards is still required). This accessory is optional. You don’t have to use them, but look how cool they make your face-down creatures look! They're also a good way to remember to show your opponent what your face-down creature was if it leaves the battlefield or the game ends.
