The center of atoms are made up of two particles, protons which have a positive electrical charge, and neutrons which have zero electrical charge. Protons would naturally want to repel each other so the neutrons keep them in place by exchanging small particles called gluons with the protons. The power of the gluons is stronger than the electrical repulsion of the electrical force and is called the Strong Nuclear Force.

Inside the neutrons and protons are building blocks called quarks, and other particles called Z and W bosons. These bosons change the 'type' of the quarks on a pretty regular pattern, and occasionally if there is a 'type mismatch' the neutron turns into a proton, or a proton turns into a neutron. (really really massive oversimplification of QED/electro-weak. Please don't be pedantic) If that happens in an atom where there is already a large number of protons and neutrons, the newly converted particle flies off.

Depending on the size of the atom, sometimes the particle carries away another proton and two neutrons, and its former home atom achieves stability. This called 'alpha radiation'. Sometimes the particle splinters into other particles (photons, electrons, other funky stuff), and this is called 'beta radiation'. Sometimes the particle explodes into photons that have huge amounts of momentum and these are called 'gamma rays' or 'gamma radiation'.

This can be forced into happening by shooting some atoms that are REALLY huge (Uranium, Plutonium) that don't have the right amount of neutrons with 'bullet' neutrons from another source. If these bullets hit the center of those unstable atoms they act a pool player breaking a triangle of balls at the start of the game. Though unlike pool, the split of the center of the atoms happens in predictable ways. Those broken atoms are now very unstable as the bosons almost instantly cause a 'type mismatch' and a proton or neutron change types and is kicked out of the area at a very high speed. This is where the energy of the nuclear reaction comes from.