When an unstable nucleus emits radiation, its mass number, its atomic number, or both can change. The atomic number is the number of protons, so it also gives the charge of the nucleus.
In alpha decay the nucleus loses two protons and two neutrons. Its mass number falls by 4 and its atomic number falls by 2, so both its mass and its charge decrease and it becomes a different element. In beta decay a neutron in the nucleus changes into a proton and an electron, and the electron is emitted. The number of particles in the nucleus stays the same, so the mass number is unchanged, but there is one more proton, so the atomic number goes up by 1.
Gamma emission gives out energy only. The mass number and the atomic number are both unchanged, so the nucleus keeps the same mass and charge. Emitting a neutron lowers the mass number by 1 but leaves the atomic number the same, so the nucleus is a different isotope of the same element.
