Z: The atomic number, the amount of protons in the nucleus (also describes the chemical element)
A: The mass number, the total amount of nucleons (protons and neutrons) in the nucleus.
N: The number of neutrons in the nucleus (N = A-Z)
Nuclei with the same Z but different N are called isotopes.
Nuclei with the same N but diffetent Z are called isotones.
Nuclei with the same A are called isobars.
Most nuclei have unstable or radioactive isotopes. Radioactive isotopes can decay via multiple channels. The most well known decay methods are alpha decay (emission of a Helium-4 nucleus), beta decay/electron capture and gamma decay (high energy photon). Half-lifes can range from fractions of a second to millions of years.
Nuclei also have a ground state and multiple excited states. Each of these states is characterized by a spin and a parity. In some cases the nuclear Shell Model can be used to predict the spin and parity of the ground and excited states.
The nucleus also has a magnetic dipole moment (or nuclear magnetic moment) and an electric quadrupole moment. The former is related to the magnetic dipole moment of the constituents of the nucleus (protons and neutrons). The latter is related to the charge distribution (and hence the ‘shape’) of the nucleus (spherical nuclei will have small quadrupole moment, more deformed nuclei will have larger quadrupole moment).