In The Vampire Diaries vampires don't breathe but whenever Stefan or Damon kill another vampire it's usually by taking out their heart. Why do they need their heart if they don't breathe? They're dead so how would their heart even work?
The show isn't 100% consistent, but in this show vampires seem to continue most bodily functions. They can eat (and even enjoy) regular food. They can get drunk from alcohol (but it takes much more), they can have sex, etc.
Since they can speak, they can obviously still move air over their vocal cords, so the lungs are working.
The assumption then is that the heart is still pumping to carry blood to every cell, just like a human.
In Season 7
Caroline is pregnant with Alaric's twins, he explains that since her heart is still pumping there is no reason that her body technically can't support a fetus.
A broken neck "kills" them, but then they heal and revive. But it can take several hours before they come back. So they can suffer severe spinal cord damage.
When Stefan is trapped in the safe he is shown repeatedly drowning due to lack of air, then reviving and 'dying' again.
The vampires in this show are the result of a magical spell. The effect of this spell is such that as long as consume blood (doesn't even have to be human), they will not age and can heal from nearly any wound. But otherwise they go on "living" just like they did as humans. And since the witches hand out daylight rings like candy, its really business as usual.
Removing the heart, decapitation, and presumably fire are the only things that they can't fully heal from.
DISCLAIMER: I have found no specific canon explanation for the heart-removal thing. Adding to this, vampire lore has roots in many different and varied cultures, many of whom disagree about how to kill a vampire, so there will be a lot of possible answers to this.
We humans, with our functional organs and souls generate vital (read: living) energy as a natural process of sustaining our lives; we eat and drink, we rest, we have sex, we express ourselves artistically, we sustain our spirits with esoteric and/or religious pursuits, and so on.
As an undead, vampires don't generate this vital energy themselves, and thus must obtain it in other ways. Vampires do so through the consumption of the vital blood of human (and sometimes animal) victims. Our living blood is what is renewing their dying flesh, which allows them to continue their existence. In this way, the heart isn't much more important than the lungs; neither are functional or carrying on the process of living.
In The Vampire Diaries universe, the focus of this vital essence is the heart. Remove it, and you remove what little vital essence ties the vampire to its undying body. As the show based on the series seems to treat the heart as the "focus" of the curse, this would also make sense from another perspective: by removing the focus of the curse, the curse is broken, and the vampire simply becomes a run-of-the-mill corpse.
In the swarm of vampire folklore tales which eventually conformed the myth, there are a branch (mainly from the east-european regions) where the heart of the creature is it's weak point.
The ways to kill the beast differs from tale to tale, on several the vampire has to be stabbed with a blessed wooden stake, while in others the material of the stake doesn't bothers at all, and others where the heart can be destroyed in any way (even extracting it from the corpse), there are much of them in which further security measures must be taken to utterly destroy the vampire: chopping his head, filling the mouth with blessed host or garlic, burning the corpse and the head and bury them separately being the most usual, but not the only ones.
So, in this specific show, probably the writters decided to take roots on some of the original folklore, defining a clear way to kill the vampires in their universe that makes easy and clear when a character is dead. This way, they can focus on other plots, and they don't have to speculate with the efficiency of different killing ways.