Beta decay is a type of radioactive decay involving the "weak force" and the transmutation of nucleons (ultimately, the transmutation of quarks) inside an atomic nucleus. It is one of three major types of radioactivity (the other two being alpha decay and gamma decay), and the only one that involves the transmutation of a subatomic particle. The most common radioactive decay involving the weak interaction is the transmutation of a neutron into a proton, with emission of an electron and an antineutrino.
Before the structure of the nucleus was understood, this emission of negative particles was observed, and they were called "beta particles" or "beta rays". The beta particles were later determined to be just electrons. At this time, it was believed that the electrons resided in the nucleus all along, and somehow escaped. That is, it was believed that, for example, Potassium-40 had 40 protons and 21 electrons in its nucleus, in addition to the 19 electrons circling outside of the nucleus. This gave it a net positive nuclear charge of 19. When it underwent beta decay, one of the nuclear electrons escaped, resulting in a Calcium-40 nucleus with 40 protons and 20 electrons, and a net positive nuclear charge of 20. We now know that Potassium-40 has 19 protons and 21 neutrons. When it undergoes beta decay, one of the neutrons turns into a proton, an electron, and an antineutrino. The resulting Calcium-40 nucleus has 20 protons and 20 neutrons. Since the neutron and proton have nearly equal mass, and the mass of the electron is negligible in comparison, either model was consistent with observed nuclear charges and atomic weights. It took the development of quantum mechanics, and the discovery of the neutron, to realize that the former model was incorrect.
Under the quark model, the transmutation of a neutron into a proton is actually a transmutation of one of its constituent quarks. The neutron consists of an "up" quark and two "down" quarks. The proton consists of two "up" quarks and one "down" quark. The "up" quark has an electric charge of +2/3, while the "down" quark has a charge of -1/3. During beta decay, one of the "down" quarks turns into a "up" quark, with the extremely short-lived "weak boson" W- carrying off the charge difference of -1. That boson quickly decays into an electron and an antineutrino.
There are other, much rarer, radioactive decays involving the weak interaction. One is the "beta-plus" decay, in which a proton turns into a neutron, emitting a positron (antielectron) and an electron neutrino. This action, by itself, is forbidden by conservation of energy, so it can only occur in nuclei that have unusual energy configurations. Another is the "electron capture" event, in which a proton captures one of the electrons circling the nucleus, turns into a neutron, and emits an electron neutrino. While this is not energetically unbalanced, it requires that an electron get very close to the nucleus, which is extremely rare under the rules of quantum mechanics.