As another answer here has already clarified, an atom’s central nucleus consists of two kinds of particles: protons, and neutrons.
You can think of a proton as having two little magnets inside of it. (These aren’t literal magnets you’re used to, they’re special metaphorical magnets.) We’ll call one the weak magnet and the other the strong magnet. Each magnet behaves independently and only reacts to other magnets of its type.
The weak magnet is, as its name suggests, somewhat weak. It’s no slouch, but compared to the other magnet, it’s weaker. You can feel the effects of this magnet very, very far away from the source. All of these magnets repulse one another.
The other magnet is, again as its name suggests, quite strong. Way stronger than the other one. But its effects can only can be felt at extremely short distances (like, only a couple proton’s widths away). All of these magnets attract one another.
As stated, the proton has both the weak and strong magnet inside of it. Neutrons have the strong magnet, but not the weak magnet.
The strong magnet can overpower the weak magnet, but only when the two strong magnets get within close range. Otherwise, the weak magnets dominate. This means that atomic nuclei (which are clusters of weak-magnet bearing protons and some neutrons) normally repulse one another, until you get them close enough together to basically touch, in which case, the strong magnet takes over and they’ll snap together.
Generally, for fusion reactions, this “snap together” part of the reaction is where the energy comes from. That violent snapping causes the newly formed atom to start zipping around faster than the parts that made it up were going before, which can be harvested as useful energy.
One small wrinkle to this explanation: the specific case of two hydrogen atoms snapping together isn’t actually a thing that creates any energy. For reasons that are well over both of our heads, two protons just really don’t like being stuck together alone. To stay together, they need some neutron friends around. I guess you could tongue-in-cheek say that they need some moral support? The actual energy-generating step in this process is when a proton and neutron, already bound together, find a third proton and snap together. From there, these new Helium-3 atoms [have a couple options](https://en.wikipedia.org/wiki/Proton%E2%80%93proton_chain#The_proton%E2%80%93proton_chain) in what path they take to eventually add an extra neutron to the bundle and create Helium-4 atoms, releasing even more energy.
As the linked article says (if you can follow, not ELI5 but there are some pictures), the initial proton-neutron pair I mentioned earlier is created when two protons come together, stick together, and one of the protons “transitions” to a neutron. They can do that, but the conditions for it are really, *really* rare. In a thick, thick nutshell, it has a chance to happen when two protons get, for lack of a better word, “stressed out”, such in the case where two protons are forced to be really close to one another. This process doesn’t create any notable useful extractable energy, which is why I said that the two protons case isn’t the energy-producing step.
By the way, the “weak magnet” in my explanation is the electromagnetic force. Notably, the force responsible for creating the magnetism you know, but in this case we’re talking about electric charge instead of actual magnetism. It’s the positive charge that all protons have.
The other “strong magnet” in my explanation is a completely different force of nature called… the strong force. No, really. It actually gets its name for the same exact reason I gave it that name in the example–it’s a lot stronger than the electromagnetic force. That’s literally it. Its main effect is keeping quarks (even smaller particles that protons and neutrons are made of) held together to make protons and neutrons in the first place. But a sort of “side-effect” of that is the “strong magnet” effect I described earlier that causes the protons and neutrons to stick to each other. Kind of like… I dunno, popcorn balls, or rice crispy treats. Individual little bits that you bind together with sticky marshmallow filling, and compact into individual balls or bars. But the treats are still a bit sticky and will probably stick to one another if you let them touch.
The universe has four fundamental forces we currently know of: gravity, electromagnetism, the boringly-named strong force, and the even more boringly-named weak force. The weak force gets its name because it’s so, well, weak. Way weaker than the other three. It’s responsible for a lot of weird effects that cause particles to transform into other particles, including that “protons becomes a neutron” business I mentioned earlier. The weak force’s, uh, *weakness* is why that process is extremely rare.
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