It's funny how things like this go crazy when you look at how different editions on different languages work.
Hit Dice, in Portuguese, was translated (incorrectly) as "Dado de Vida", or "Life Dice". In portuguese, you don't have "Hit Points", you have "Pontos de Vida", or "Life Points".
So, at least for portuguese-speaking people, associating "Life Points" with "Healing"(Gaining "more life") is trivial.
The same don't happen on english, however. I had a player who was on the same situation a few days ago, having this same issue (albeit on another system). He simply couldn't plug on his mind that the same value for "Hit Dice" would be used for Healing.
So, we went up to try to find a few mnemonic ways to remember that easily. We came up with something... fuzzy. We went to write some verses, to find something easy to remember. We came up with this (Keep in mind that I'm a GM, not a Poet, so yeah, that's a heck of a silly thing):
When Ones roll plenty, and the battle goes bad,
No fear I have, I have to say that,
One of them is a Cleric, and good one in fact,
Always prompt, ready to act
And because of him, I always remember
That to my well being, on that dreaded chamber
Dear to the Heart, My hit dice should I Keep
'Cause with them he will heal me, in time of need
(It's actually bigger than that, but I removed the non-relevant parts)
The last two verses did the trick, when you read them in a "Dr. Seuss"-like way.
Yeah, it's bad, but hey, it worked!
According to the Basic PDF (page 74)
A creature's current hit points (usually just called hit points) can be any number from the creature's hit point maximum down to 0.
So no, negative hit points do not exist in 5e. The archived development-poll blog post "A Close Call with Negative Hit Points" explains the history of negative hit points and part of the reasoning for the change in 5e.
Best Answer
When Hit Points are Reduced to Zero, what happens?
There are no negative hit points in DnD 5e.
Any damage that is equal to or greater than the amount of hit points you have remaining will set your hit points to 0. (Basic Rules p. 79).
Hit points decrease or increase based on damage taken and healing received, respectively.
Any skill or spell that restores hit points to you (heals you) will work if you are at 0 hit points and not dead.
As soon as you are at 1 HP or greater, you are conscious and able to function.
If you are reduced to 0 Hit Points, are you dead?
Not necessarily, but you may suffer instant death.
If instant death does not happen, the character at 0 HP is unconscious. (Basic Rules p. 79).
I'm unconscious at 0 HP. Now what?
You roll death saving throws each turn, which will result in you being stabilized and unconscious (3 successes) or dead (3 failures). During this time someone or something else can either stabilize you or heal you.
If you are stabilized, you remain unconscious. (PHB p. 79). (Hopefully, your other party members will help out).
While unconscious, if you are stabilized then death saving throws are no longer necessary.
If you are healed to 1 HP or greater during that time, you are alive, awake, and no longer in the 0 HP condition.
Note: if you roll a 20 on a d20 for a death saving throw, you are now conscious and have 1 HP. (Basic Rules p. 79).
How to do I get my hit points back?
All of the above are spelled out in the free Basic Rules download, and in the Players Handbook.
1. A character must have at least 1 hit point when starting a long rest to gain any benefit from it.