How does fusion power work?

I'm just looking for a simplified version of it, so that I can understand.

Answer:
Basically, inside the atom there are protons, neutrons, and electrons. Protons and neutrons are bound together in the nucleus, while electrons float around outside them.

It takes an enormous amount of energy to hold them all together and prevent them from just repelling and falling apart. This energy is really huge, like signicantly more than colliding cars together.

So fusion involves smashing two or more atoms together at close to the speed of light, which in turns breaks the bonds holding them, releasing that huge energy. The energy can then (theoretically) be harnessed and put to use. Because its so much energy though, we have a hard time containing it, its like keeping a nuclear explosion inside a jar.

Also, the second responder has it backwards. The sun fuses the hydrogen atoms into helium, not helium into hydrogen.
In fusion two or more smaller atoms join together to form a single atom of higher atomic weiht and the process generates heat energy.. Two helium atoms joining to form one hydrgen atom as is the case in the sun, is a fusion phenomenon.
Fusion power relies on the mass / energy relationship developed by Einstein:

E = mc^2

When two hydrogen nuclei (actually usually two nuclei of deuterium, an isotope of hydrogen containing one proton and one neutron) are fused together, they will form a helium nucleus. This fusing takes an incredible amount of energy to overcome the natural magnetic repulsion of the two nuclei due to their +2 charges. However, if one can overcome that energy platform, the resultant helium nucleus is actually more stable (requires less energy to keep itself from flying apart) than the two deuterium nuclei that formed it. This energy can actually be measured as mass, as the atomic mass of the deuterium nucleus is actually slightly less than 1/2 the atomic mass of the helium nucleus. So the resultant nucleus weighs less than the parts that formed it (because less energy is required to maintain it), and this excess mass is converted to free energy which is released. The energy released far exceeds the energy required to bring the two nuclei together, so the reaction can be self sustaining.

As a previous answerer has noted, the problem is performing this all in a controlled manner, and collecting and removing the excess energy in a usable fashion. That's a problem that will have to be solved before fusion power can be considered a practical source of power for anything other than bombs.

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