Technically speaking, alpha and beta decay are both types of nuclear fission.  Fission is the breaking down of an atom's nucleus into smaller parts.  Specifically, alpha decay is the breaking down of a nucleus to produce an alpha particle (a helium atom consisting of two protons and two neutrons).  This produces an element that is two protons smaller than the parent atom.  Beta decay is the breaking down of a nucleus to produce a beta particle (high energy electron).  They do differ somewhat in their nature, however.  Both alpha and beta decay are natural and spontaneous processes in heavy, radioactive elements.  They produce predictable products.  Fission, on the other hand, is usually induced by bombarding atoms with high energy neutrons to catalyze nuclear decay.  The resulting products are often varied and unpredictable.