As you know, an atom consists of at least one proton, and at least one electron. Protons are positively charged, electrons are negatively charged.
Protons repel each other, which is called the electrostatic force. But there is another force called the strong nuclear force - that acts when protons are very close together, and it makes them stick together. That's how elements can have as many as 110 protons stuck together in their nucleus, even though protons repel each other.
So nuclear fusion is the process of taking two atoms, and bringing them together with enough energy that you can get over the electrostatic force keeping their nuclei apart. If you can do this, and get the nuclei to collide with enough energy, the strong nuclear force takes over and you get a new atom.
We have successfully done this with hydrogen. If you smash together an atom of deuterium (which is hydrogen with one neutron) with an atom of tritium (hydrogen with two neutrons,) you get a standard atom of helium - 2 neutrons and 2 protons. You also get a shitload of energy, because your helium atom has less mass than the two hydrogen atoms, and E = mc2
The problem of course is that you need a ton of energy to get fusion to happen. So the first way that humans figured out how to get this "ton of energy" was by first detonating an atomic bomb. Detonate an atomic bomb, create a bunch of energy, use the energy to force hydrogen to undergo nuclear fusion, and now you generate a bunch more energy. We call this a fusion bomb, or a hydrogen bomb.
As to how we harness the energy of a fusion reaction in a way that we can use it to generate electricity... well, we haven't figured that out yet. There are some promising theories, but nothing that's been implemented at the power plant scale to date.