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.