The strong force or strong interaction is one of the fundamental forces of the universe, in this case, the force that holds the nucleus of atoms together. In particular the interactions between protons and neutrons in elements (these are also called nucleons)
The nucleus of an element, this is where the strong force or interaction takes place. The red nucleons represent protons, the blue represent neutrons. image: wikipedia
The image shown above is really just a model of the nucleus of an atom. In reality the protons and neutrons likely move around each other in different ways, but still tightly bound.
Each nucleon in turn is made up of smaller particles called quarks. Protons have different types of quarks than neutrons do, leading to different properties, particularly differences in charges. Protons being positively charge and neutrons having no charge. Having a charge in this case just means if it will be affected by a magnetic field.
How it works, approximately...
The strong force is mediated by particles called gluons. These particles cause quarks to be produced if quarks start to separate too far from each other. The range of this interaction is very short but also really strong. It needs to be strong in order to hold the nucleus of atoms together. Sometimes atoms with awkward numbers of neutrons usually, the atom can become a bit unstable and hence radioactive.