Atomic power, also called nuclear power, can come from two distinct types of reactions between atoms: fusion and fission. To understand these reactions, you must first know the basic structure of an atom. An atom consists of a positively charged nucleus in the center surrounded by negatively charged particles called electrons. The nucleus is made up of positively charged particles called protons and non-charged/neutral particles called neutrons. The number of protons and neutrons an atom has determines what kind of element it is, such as gold, iron, carbon, etc., and different elements combine to form different chemicals.

The electrons are attracted to the nucleus by a force known as the electrostatic force (i.e. negative charges attracted to positive charges).The protons and neutrons in the nucleus are held together by a much stronger force called the nuclear force, which happens only when particles are very close together (10-15 meters). Because the bond between the protons and neutrons is so strong, a lot of energy can be released when those bonds are broken.

Fusion occurs when nuclei from two atoms fuse to form one single atom. Fusion does not occur under standard conditions because the atoms must overcome the electrostatic repulsion that occurs when the two positively charged nuclei come together. When the atoms are forced together and come close enough to one another, the nuclear force overcomes the electrostatic repulsion and allows the two nuclei to become one.

In fission, one heavy atom splits to become two smaller atoms. Fission occurs because heavier atoms have greater electrostatic repulsion (i.e. more positively charged protons in close proximity to one another) than smaller particles. With enough positive charges, the electrostatic force overcomes the strong nuclear force and the large atom fragments.

In both cases, bonds between protons and neutrons are broken and more than a million times more energy is released than in average chemical reactions. Energy from fusion has been harnessed for thermonuclear weapons and fission has produced the atomic bomb. However, nuclear power can and is used peacefully within nuclear reactors to produce clean energy. Fission is currently used at nuclear power plants, but future power plants may use fusion instead.

If you would like to know more, there are some great resources online. One website that I like is this:

Atomic power (also called nuclear power) uses nuclear fission to generate electricity. Nuclear fission occurs when the nucleus of an atom is split into two pieces. This process releases a lot of heat (as well as other types of radiation including neutrons). In a nuclear reactor, a chain reaction occurs. This means that when one atom is split, it causes other atoms to split too. The heat produced by atoms splitting in a nuclear reactor is used to boil water. The steam from the water is used to turn a turbine and generate electricity. Coal, oil, and gas power plants also use heat to boil water and turn turbines, just like in a nuclear power plant. The major difference is the source of the heat.

Atomic power relates to chemistry in a lot of ways. Nuclear fission only occurs in atoms with large, unstable nuclei like Uranium-235 and Plutonium-239 (these atoms are called "fissile" because they can undergo fission). This relates to a field called nuclear chemistry where radioactive elements are studied. Refining these elements from ores, understanding the properties of the materials used to build safe nuclear reactors, and even boiling water are all related to chemistry.

Atomic energy really isn't chemical at all. Chemistry is determined by the interactions between electrons in the outer part of an atom with those of other atoms. Atomic energy involves changes in the state of the atomic nucleus. They're two, very different things.