Nucleus
The nucleus of an atom contains protons and neutrons -- positively charged particles and neutral particles. The particles are so close together that the strong nuclear binding force overrides the electromagnetic repulsion associated with like-charged particles. The nucleus of an atom makes up the majority of the mass; but it is a very small fraction of the space. The number of protons in the nucleus -- which represents the atomic number -- determines what element the atom is. A neon atom will always have ten protons in its nucleus; changing this number changes the element.
Electron Cloud
The electron cloud makes up the negatively-charged part of the atom, the part responsible for most of its chemical interactions with other atoms; due to electron movement from one atom to another. Middle-school science classes often describe the electrons rotating around the nucleus as resembling the planets orbiting around the sun, but this is false. Electrons have a chaotic movement pattern and move at the speed of light, so there is no way to track their exact location and velocity at any particular instant. Therefore, the electrons are said to move in a cloud around the atomic nucleus.
Nuclear Reactions
There are two types of nuclear reaction: fission and fusion. A fission reaction, such as the atomic bombs dropped on Hiroshima and Nagasaki, occur when a stray neutron strikes the nucleus of an atom with enough force to disrupt the bonds holding the nucleus together, causing it to split and release this binding force as energy.
Fusion, on the other hand, comes from the combination of smaller nuclei; such as the hydrogen-helium fusion that occurs in the sun.
Ions and Isotopes
On the macroscopic level, most matter is neutrally-charged overall; it has the same number of protons and electrons. Atoms that have more or less electrons are called "ions." They have a negative or positive electrical charge, and are attracted to the opposite charge. This is the basic reason for ionic and covalent bonds between atoms.
Isotopes of an element have the same number of protons, but different numbers of neutrons. This affects the weight and stability of the atomic nucleus, causing it to emit radiation.