Protons exist in the nucleus, which is the dense core of an atom. Along with neutrons, protons make up the vast majority of the mass of an atom. Electrons orbit around the protons and neutrons, primarily because protons are much denser than electrons. However, while the protons and neutrons are tightly packed together in the nucleus, there is a relatively huge amount of empty space between the orbits of the electrons and the nucleus.
The discovery of the proton is generally credited to Ernest Rutherford, who oversaw the famous "Rutherford Scattering" experiment in 1909 (which was actually conducted by Hans Geiger and Ernest Marsden, with the results interpreted by Rutherford). The experiment, which involved shooting alpha radiation particles at gold foil, proved that atoms had densely-packed nuclei, which had to be made up of even smaller particles. Rutherford coined the actual term "proton" in a 1920 publication.
Protons are the most important feature to understanding the characteristics of an atomic element. Atoms may change their number of electrons and still maintain their identity as that atom; this is known as an ion. Indeed, elements frequently fluctuate in their number of electrons when combining to form molecules or compounds. However, if the number of protons in an atom changes, the fundamental qualities of the atom changes and it goes from being one type of element to another (e.g., adding one proton to hydrogen causes it to become helium). The process of adding protons to an atom is known as "nuclear fission" and the splitting of protons is known as "nuclear fusion."
Despite their small size, protons are not an elementary particle, and are themselves made up of even smaller particles known as "quarks," which also are the building blocks of neutrons. Quarks are considered "elementary particles," meaning that there is nothing that is smaller than a quark. There are six types of quarks: up, down, charm, strange, top and bottom. Protons and neutrons are only made out of up and down quarks, and they are part of a group of quarks known as "baryons." Particles made from other types of quarks are known as "mesons."