Though the positively charged protons exert a repulsive electromagnetic force on each other, the distances between nuclear particles are small enough that the strong interaction (which is stronger than the electromagnetic force but decreases more rapidly with distance) predominates. (The gravitational attraction is negligible, being a factor 1036 weaker than this electromagnetic repulsion.)
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The discovery of the electron was the first indication that the atom had internal structure. This structure was initially imagined according to the "raisin cookie" or "plum pudding" model, in which the small, negatively charged electrons were embedded in a large sphere containing all the positive charge. Ernest_Rutherford and Marsden, however, discovered in 1911 that alpha particles from a radium source were sometimes scattered backwards from a gold foil, which led to the acceptance of a planetary model, in which the electrons orbited a tiny nucleus in the same way that the planets orbit the sun.
A heavy nucleus can contain hundreds of nucleons (neutrons and protons), which means that to some approximation it can be treated as a classical system, rather than a quantum-mechanical one. In the resulting liquid-drop model, the nucleus has an energy which arises partly from surface tension and partly from electrical repulsion of the protons. The liquid-drop model is able to reproduce many features of nuclei, including the general trend of binding energy with respect to mass number, as well as the phenomenon of nuclear fission.
Superimposed on this classical picture, however, are quantum-mechanical effects, which can be described using the nuclear shell model, developed in large part by Maria Goeppert-Mayer. Nuclei with certain numbers of neutrons and protons (the magic numbers 2, 8, 20, 50, 82, 126, ...) are particularly stable, because their shells are filled.
Since some nuclei are more stable than others, it follows that energy can be released by nuclear reactions. The sun is powered by nuclear fusion, in which two nuclei collide and merge to form a larger nucleus. The opposite process is fission, which powers nuclear power plants. Because the binding energy per nucleon is at a maximum for medium-mass nuclei (around iron), energy is released either by fusing light nuclei or by fissioning heavier ones. Helium was formed from hydrogen in nuclear reactions shortly after the Big Bang, while all the other elements in the universe were created in supernova explosions.
Nuclear reactions also occur naturally on earth. Temperatures and pressures here are not great enough to overcome the electrical repulsion between nuclei and allow fusion, but heavy nuclei such as uranium may undergo fission and alpha decay, and beta decay can also occur. Alpha decay can be considered as an extremely asymmetric case of fission, in which one fragment is a helium nucleus (alpha particle). In beta decay, either a proton is converted into a neutron (with the emission of an antielectron and a neutrino) or a neutron is converted into a proton (emitting an electron and an antineutrino).
Much of current research in nuclear physics relates to the study of nuclei under extreme conditions. The heaviest of all nuclei are neutron stars. Nuclei may also be characterized by extreme shapes (like footballs) or by extreme neutron-to-proton ratios. Experimenters can also use artificially induced fusion at high energies to create nuclei at very high temperatures, and there are signs that these experiments have produced a phase transition from normal nuclear matter to a new state, the quark-gluon plasma, in which the quarks mingle with one another, rather than being segregated in triplets as neutrons and protons.