What part of the atom is involved when an atom decays?

Really, it is the whole thing.

An atom decays because the nucleus is unstable. The nucleus contains protons and neutrons, and only certain combinations of the numbers of each are stable.

When a radioactive atom decays, it can do a number of different things. Best known by ordinary people, possibly, is nuclear fission, in which the whole atom breaks apart to become two atoms. In the process various pieces of the atom also are released.

Sometimes an atom decays by emitting an alpha particle, which means that two neutrons and two protons, combined in a single package, are emitted. This also implies that two electrons are somehow lost. In the process, the atom changes to a different element, with an atomic number reduced by two, and with an isotope number reduced by four.

Sometimes an atom undergoes a beta-negative emission. This means that the number of electrons and protons in the atom is increased by one, but the number of neutrons is reduced.

Sometimes an orbital electron is captured, to combine with a proton in the nucleus to make a neutron. When this happens, the atomic number is reduced by one, so the atom is of a different element, but the isotope remains the same.

There is a type of decay called an "isomeric transition," in which an atom emits a gamma ray (type of photon) but keeps both its atomic number and its isotope number. The notation of the isotope number before the has an "m" attached (indicating an excited meta state), which is gone after the transition has taken place. For example, zinc-69m decays to zinc-69.

These are just examples illustrating how the whole atom is affected. There are a number of others.