Strong Force
Current models of physics describe four fundamental forces present throughout nature: the electromagnetic force, the weak force, the strong force and gravity; the strong force is the strongest of the four. Unlike the others, however, this force is also short range, meaning the particles must be separated by only an extremely small distance before they experience its effects. Within this short range, the strong force is indeed quite strong --- powerful enough to overcome the repulsion among the positive charges of the protons.
Distances
Physicists describe the strong force as an interaction mediated by exchange of particles called gluons. For the strong force to overcome repulsion, the distance between the particles must be roughly equal to the diameter of a proton or a neutron. At greater distances, the repulsion between positively charged protons makes them fly away from each other. Neutrons also experience strong-force interactions, but they don't experience electromagnetic repulsion because they have no charge.
Fusion
The short range of the strong force explains why nuclear fusion, the process that takes place in the sun's core, requires extremely high temperatures to occur. Atomic nuclei don't ordinarily come close to each other, because the repulsion between positive charges holds them apart. To overcome that repulsion, the nuclei must be moving extremely fast --- so fast that the repulsion is insufficient to stop them. Once they collide, the strong force holds the particles together.
Neutrons
Bombarding nuclei with neutrons, by contrast, is much easier, because the neutrons don't experience any electromagnetic repulsion, and if they come close enough to a nucleus, they can become captured by it. Neutrons are also important for the stability of larger nuclei. By holding the protons apart, they help to decrease the repulsion among the positive charges, and they also increase the strength of the forces binding the nucleus together.