I played Magic: The Gathering when Unlimited first came out, and quit shortly after Antiquities were released. About a month ago I started playing again.
I was trying to find what the biggest changes are, and saw someone say the biggest difference was the elimination of Damage on the Stack. I've read the new rules and noticed a few differences, but am not sure what "Damage on the Stack" refers to as being different. Could someone explain what it was, and how it has been eliminated?
Formerly, in combat, it would go something like this:
Declare Blockers -> (Players can do stuff) -> Stack Damage -> (Players can do stuff) -> Damage resolves
This allowed players to do various combat "tricks." Some examples: You attack with a 2/2, and I block with my Mogg Fanatic. After damage was stacked, I could use Mogg Fanatic's activated ability, sacrificing him and dealing an additional 1 damage to your 2/2. This 1 damage, coupled with the 1 combat damage on the stack, would kill your 2/2.
Under the new rules, this is not possible (and thus Mogg Fanatic has been significantly weakened). I could either deal the 1 combat damage to your 2/2, OR sacrifice him to deal 1 damage to any target. By sacrificing him, he no longer is alive to deal the combat damage; and I can't sacrifice him after damage because he would already be dead. From a "common sense" perspective, this seems more intuitive, and I think that's why they changed the rules. (Personally I liked Damage on the Stack better; it added more possibilities).
Here's another example, just to be perfectly clear: Suppose you attack with a 2/2 and I block with a 2/2. Formerly, I could stack the combat damage, and then use a "bounce" card such as Unsummon or Boomerang to return my guy to my hand. The damage would then resolve; your guy would die, and mine would safely be in my hand.
With the new rules, I can block your guy (preventing damage to me) and still bounce my guy to save him from death, but I do not get to also kill your guy. Thus, bounce effects have also been weakened.
If you haven't played since Antiquities, you're probably unfamiliar with the Stack in general -- it's a pretty important part of how Magic works now, and you should familiarize yourself with it. Basically it's a Last-In, First-Out stack that spells and abilities go into before they resolve. Once both players pass priority in succession without adding anything to the stack, the top effect happens and the active player gets priority again. (The concepts of priority and the active player are also pretty important -- it's what allows you to do things like use one of your Planeswalker's abilities before they have a chance to Bolt it or whatever).
Damage on the stack was never a "phase" as LittleBobbyTables said; it was something that happened during the combat damage step of the Combat Phase. Creature damage was treated like ever other ability, used the stack, and could be responded to. With the rules overhaul that happened around M10's release, this was changed, as it led to too many unpleasant situations of new players getting taken advantage of by more experienced players, and because combat without it is actually more tactically interesting -- you now have to choose between your creature's power and his sac ability, rather than always getting both. It does unfortunately render a lot of older cards terrible, but they have definitely started pushing the power level of sac effects on newer cards to compensate.
Mana burn was also eliminated in that rules update, so be aware of that too.
"Damage on the stack" was a phase after damage had been assigned, but before actually applied, where you could do fast effects and all sorts of fun stuff.
For example, you could block a Grizzly Bears with a Prodigal Sorcerer, then with "damage on the stack", tap the Prodigal Sorcerer to do a point of damage to the Grizzly Bears.
The Grizzly Bears would now have a point of damage on them, and then when damage resolves, the second point of damage (from the Prodigal Sorcerer's 1 point of power) would kill it.
Basically, you could either attack or block, then use a creature's effects for added bonus. Mogg Fanatic and Nantuko Husk were fun for this, but there were plenty of other creatures that abused or took full advantage of "damage on the stack".
You'll also want to know that once an ability goes on the stack, it still resolves (unless the ability is countered) even if the source is no longer around. For example, the old question was if a Prodigal Sorcerer taps to deal one damage to a Royal Assassin, then the Royal Assassin responds to kill the now tapped Prodigal Sorcerer, which creature dies? It used to be the Prodigal Sorcerer would die, because it was no longer around to deal the damage, but now they would kill each other.
