The weak force is the force that is responsible for some forms of radioactivity. It is one of the four fundamental forces; the others are electromagnetism, gravity and the strong force. The weak force and the strong force only act on a very short range, within the nucleus of an atom. Of these two nuclear forces, the weak force is the weaker, hence the name.
Some radioactive materials undergo a process called beta decay, in which a proton in the nucleus is converted in a neutron, or a neutron in a proton. However, a proton has a positive electric charge while a neutron has no charge. So that the law of conservation of electric charge is not violated, another particle with a positive charge (a positron) is created when a proton changes in a neutron. Positrons are also called beta particles, and hence the name beta decay. In fact, there is another particle which is created together with the positron; this is the elusive neutrino.
On a more fundamental level, both a proton and a neutron consist of three quarks. In the case of a proton, these are two up quarks and one down quark, while a neutron consists of two down quarks and one up quark. The weak force changes a proton in a neutron by changing an up quark to a down quark and creating another particle, the W boson. This W boson decays in the positron and neutrino which are emitted during beta decay.