From D&D 3.5 to 5.0(5E) - 6 - Death and Dying in 5th Edition Rules

Dying in 5.0 will not be pleasant.  Instead of the slow drift toward -10, there is a stressful set of die rolls waiting to bring the grim reaper in an instant.

One way to die in 5.0 is via Instant Death.  Now this is not much different than in 3.5.  If your character takes an amount of damage that exceeds her current HP + her maximum HP, she is immediately dead.  So if your cleric with a maximum of 8 hit points currently has 1 hit point and she takes 9 or more hit points in a single attack, she is dead.
The more traditional way to die is to fall unconscious first.  If you are damaged to 0 or fewer hit points, but not enough to result in "Instant Death", then you are unconscious.  From then on, when it is your turn, you roll the d20.  If you roll 10 or higher, that is good news for you because you live to roll again on your next turn.  If you manage to roll 10 or higher on three death saving throws (that is what these are called), then you stabilize.  But if you roll below a 10, and you roll below a 10 three times before you roll a 10 or above three times, then you die.
It may sound like you have at least 3 rounds before kicking the bucket, but if you roll a 1, that counts as 2 failed rolls.  So if you have already failed once, and then you fail a second time with a roll of a 1, you're dead.  If you roll a 20 that counts as two positive rolls.
If you keep taking damage from other attacks while you are unconscious you don't need to count the HP of damage, you just count that as one more failed death saving throw.  If someone cures or stabilizes you, then you are safe from the need to continue rolling death saving throws.
Now remember, D&D is your game and you can alter the rules as you see fit.  Such customized rules that don't strictly follow the book are generally called "House Rules".  Your DM may lay down the alternate house rules for a campaign, or your entire group of players may work together to agree to some house rules.  In my group, we rotate DMs over many adventures during our campaigns, so no DM ever establishes house rules that would affect subsequent DMs or affect how players would develop their characters; we work together to agree on our house rules. 
In my group, I might recommend that we use the rules in 5.0, but that we make it 5 failed rolls before death rather than 3. 
Read it for yourself in the PH on page 197.