The atomic number is the number of protons in a nucleus. The mass number (also called the nucleon number) is the number of protons plus neutrons. Each type of radioactive decay changes these numbers in its own way.
An alpha particle is two protons and two neutrons. When a nucleus emits one, its mass number goes down by 4 and its atomic number goes down by 2. In beta-minus decay a neutron becomes a proton, so the atomic number goes up by 1 and the mass number stays the same. In beta-plus decay a proton becomes a neutron, so the atomic number goes down by 1 and the mass number stays the same.
A gamma ray is electromagnetic radiation with no mass and no charge, so gamma emission changes neither number. If a nucleus emits a neutron, the mass number goes down by 1 and the atomic number stays the same, because no protons are lost.
In beta decay the total number of nucleons stays the same, which is why the mass number is unchanged even though one kind of nucleon turns into the other.