The Sun shining from space
Fusion reactions are how the first stars got started. Soon after the Big Bang, hydrogen atoms began bumping into each other and merging into one bigger helium atom. Four atoms with four nuclei - centers - mixed together into one atom with a bigger nucleus. But each time this happened there was a bit of energy left over, and that loose energy shoots out of stars as light and heat. Inside the Sun, and other stars, millions of hydrogen atoms smash together and become helium atoms every second of every day. "Fusion" means "two things becoming one thing," and it's nuclear fusion because it's the nuclei of atoms that are fusing.
Each hydrogen atom has one electron and one proton. So four of them have four electrons and four protons. After they smash together into a helium atom, the new atom has two electrons, two protons, and two neutrons. (You make a neutron by combining an electron with a proton.) So the reassembly works out evenly; there aren't any parts left over. Where does the energy come from then? What makes the stars shine and be hot?
The four hydrogen nuclei are held together by the strong nuclear force, which acts like glue. When the nuclei are combined into one helium nucleus, it takes less glue to hold them together. That extra glue shoots away from the helium atom, mostly in the form of photons. Some of those photons reach your eyes: that's sunshine and starlight.
First hydrogen bomb explodes (1952)
People can create fusion reactions too: that's what a hydrogen bomb is. People create a nuclear fission reaction. At the center of that reaction there's a lot of heat - like in the center of the Sun - and that makes the hydrogen atoms hot enough to fuse with each other. A lot of people would like to create controlled fusion reactions, to use for electrical power, but nobody has figured out how to do it yet.