Sure, Iron Man 3 sees "genius, billionaire, playboy, philanthropist" Tony Stark remove the arc reactor from his chest, but the tech still remains at the heart of his Iron Man tech and thus at the heart of his character as well.
But other than the fact that Tony was able to miniaturize the tech in a cave "with a box of scraps," what else do you know about the amazing game-changing technology at the heart of the Earth's Best Defender? It's good for the environment. It was supposed to "shut the hippies up," as Obadiah Stane once said. Well here's a little bit more.
Howard Stark's Idea
Sure, it was Tony fans watched work with missile pieces and scrap bits in a cave to build a miniaturized version to keep pieces of shrapnel away from his heart, but it was his father Howard Stark who got the technology off the ground in the first place.
Howard was the one who designed the Arc-Reactor (some of his blueprints can be seen in Iron Man 2) and thus was able to provide the energy all of Stark Industries needed to exist. It's a family thing, building the technology to power the future.
It's All About the Electromagnet
The Arc-Reactor, whether in it's original size or its miniaturized version, is really just a power source, a batter if you will. The power, in and of itself, does not keep Tony alive once he puts it in his chest rather it powers the electromagnet so that it can keep the shrapnel out of our hero's heart.
The great thing about the arc reactor is that Tony doesn't have to worry about replacing it (like a battery) or it running out of power on him and thus allowing the electromagnet to fail and for him to die.
Stark Tower
Stark Industries is not the only brick and mortar location that benefits from the energy output of a Stark Arc-Reactor. Stark Tower in New York City, as first introduced in Avengers, is also run on Arc-Reactor energy. That means that Stark Tower, or we might call it Avengers Tower, is powered off the grid, and without any assistance from the government that the Avengers so often find themselves in conflict with.
Tony could probably make more money by offering the technology to other companies as well, but he's a little preoccupied with saving the world.
Pepper Potts Has An Arc-Reactor (in the comics)
Tony Stark is no longer the only Marvel character to have used a miniaturized Arc-Reactor to heal himself from significant bodily trauma. In an Invincible Iron Man storyline Pepper is injured by a bomb, shrapnel landing dangerously close to her heart, as it once did to Tony's.
Tony uses his technological know-how to give Pepper her own Arc-Reactor that protects her heart and, later on, powers her own super-suit, as the hero known as Rescue.
Not Threatened by an EMP
In the beginning, it was important that something, like an electromagnetic pulse, wouldn't harm the arc reactor because then, well, Tony would have died. After Iron Man 3 the arc reactor would need to work, even after an EMP, because otherwise, he might fall out of the sky or, just in general, not be very effective against villains.
The arc reactor, because of how it is designed, does not work or depends on electricity, thus the pulse of an EMP would not affect it or harm Tony/Iron Man in any way.
Wakanda Isn't the Only Place with Vibranium
In Iron Man 2 the arc reactor is poisoning Tony due to the chemical element that helps to run the arc reactor. Tony can either die through palladium poisoning or let the shrapnel drift to his heart. Of course, fans instead saw Tony use old notes and messages from his father to create a new element to run the arc reactor.
For fans who saw the film and also read the novelization, it was revealed that the element was in fact vibranium. While this was never explicitly compared with the vibranium found in Wakanda, it is an interesting easter egg. That's a pretty powerful substance to have to power the Iron Man suit.
Explosives
Why doesn't Tony want the entire world to run on clean and inexpensive arc reactor energy? If we look at Iron Man 1 and Iron Man 2 it's obvious that even a little bit of the arc reactor technology in the wrong hands can cause problems across the globe.
While the arc reactor's main purpose was to provide unlimited clean energy, it's incredible power also allows it to be used to create incredibly powerful explosive devices. With that in mind, Tony keeps most details about its creation and engineering close to the chest.
Iron Man Power
Even though Tony, in the Marvel Cinematic Universe, no longer needs the arc reactor to power the electromagnet in his chest, the arc reactor is still necessary to power every version of the Iron Man suit that Tony, or anyone else (say Pepper Potts) might use.
That's why Tony, in the later Avengers movies, always has an arc reactor on him, so that he can quickly don a suit and be ready to go. It also cuts down on the arc reactors that exist (thus not having a separate one in every single suit) and keeps them out of villainous hands.
Iron Chest Plate
The arc reactor is a relatively new invention in the timeline of Iron Man. While it has been present in the Marvel Cinematic Universe since the beginning, Iron Man's origins in the comics were a little bit different. Instead of having an arc reactor implanted in his chest to keep shrapnel away from his chest, Tony Stark was instead forced to consistently wear the chest plate from the Iron Man suit beneath his close to keep the shrapnel away from his heart.
And, unlike the arc reactor, the chest plate had to be charged every so often as well.
Superpowers without a Super-Suit
In the Iron Man comics, there have been numerous instances where the arc reactor offers its user enhanced abilities even when they are not using a super-suit of any kind (be it Iron Man, Rescue, or other).
While the arc reactor uses its power to keep shrapnel away from Tony's heart, there's obviously some power left over, and that power can be absorbed by the body.