What is the structure of the atomic nucleus?

The atomic nucleus is the center of an atom, and is composed of protons and neutrons.

Protons and neutrons are composite particles, meaning they are made of something even smaller. Both are made of quarks, which behave in ways that don't really conform to our normal expectations for particles. There are several varieties of quark, but the ones found in protons and neutrons are the "up" quark, which has a charge of +2/3, and the "down" quark, which has a charge of -1/3. 

The proton is made of two ups and a down, which mathematically works out to a charge of 1. The neutron is made of one up and two downs, which works out to a charge of 0. Their charge is the reason for their names.

An atom is defined as belonging to a particular element based on the number of protons found in the nucleus. This is primarily because the number of neutrons in the nucleus can change without affecting the atom's ability to attract electrons, and atoms regularly do end up having different numbers of neutrons with a certain number of protons. These variations in the number of neutrons are called isotopes, and they contribute to the stability of the atom; atoms that are said to be radioactive are ones that are, basically, too large for the forces holding them together, and they tend to split apart into smaller atoms in relatively predictable ways.

Despite the way it's typically depicted in diagrams as a relatively large structure, the nucleus is in fact extremely small compared to the electron orbits, and most of the atom is composed of empty space.