5e has changed Dying in a few meaningful ways from 4e's Death Saving Throw mechanic:
Death saving throws are now 3 successes before 3 failures. If you get to 3 successes, you're stable, but still unconscious. If you get to 3 failures, you die.
If you get to stable, all of your death saves are reset (successes and failures).
If any damage you would take would take you to negative your max HP, you die (though that is a single hit only, you don't track negative HP).
When you roll a 20 on your death saving throw, you become conscious and regain 1 HP.
When you roll a 1 on your death saving throw, you suffer 2 failures.
If you take damage while you are down, you suffer 1 failure. If the damage is from a critical hit, you suffer 2.
A stable dying creature regains 1 HP in 1d4 hours.
There's a little bit of a debate over whether or not getting hit while you're stable counts as a death saving throw. It definitely seems to.
That's a summary of the mechanics from 5e, it should be pretty obvious that it's a reasonably substantial deviation from 4th, though the baseline mechanic (make a d20 roll, a 20 gets you up) is the same, it does speed it up as you can't be forever dying.
At three successful saves, you stabilise and stop rolling death saves and erase any failures and the successes you've racked up. You're still unconscious, but not dying anymore, and will wake with 1 hp after some time. Erasing failures when you stabilise is so that you start the count fresh in case you de-stabilise (which can happen if you take more damage while still at 0 hp).