Atoms just aren't what they used to be. They used to be the indestructible and indivisible finite elements of matter. Nowadays they are destroyed and divided all the time. In fact, the internal structure of atoms controls how they combine to form molecules - in other words, what happens inside your atoms determines what happens in your chemistry. Things that happen inside atoms are also the tremendous source of energy inside the sun - in other words, what happens inside the sun's atoms drives all the biology on planet Earth!
In the center of the atom is the nucleus; it contains protons and neutrons. Protons and neutrons have quarks inside of them. The study of quarks is part of particle physics and is what FermiLab is all about. But that isn't what we talk about on this page! The proton and the neutron are about equal in mass and contain most all of the mass of the atom. The mass of an atom is usually given as the number of protons and neutrons in the nucleus. So for example, a carbon atom which has 6 protons and 6 neutrons is said to have a mass of 12.
Outside the nucleus are the electrons. They are much lighter (1836 times lighter) than the protons and the neutrons. The number of electrons will usually be the same as the number of the protons, in order to keep the atom neutral. They fit into specific "orbits"; two electrons can go into the "orbit" closest to the nucleus, and eight electrons can go into the next closest "orbit". The third closest "orbit" will also hold eight electrons. Figuring out why the electrons have these "orbits" is not easy!
When atoms are close to each other, electons can jump around from the
"orbits" of one nucleus to the "orbit" of another and then jump back again. Since both the nuclei are positive, they are attracted to this negative electron, and are thereby stuck together. This is the way in which atoms are stuck together to make molecules.