How does the nucleus of an atom change when it emits an alpha or beta particle?

First of all, an alpha particle consists of two protons and two neutrons (similar to a Helium atom). When an atom emits an alpha particle, its nucleus loses two protons and two neutrons. And its total mass decreases by 4 atomic mass units.(As mass of a proton or a neutron is 1 amu)

There are two kinds of beta emissions (or particles) positron or electron. 

When an atom emits a  (positron) particle, actually one if its protons releases the particle. Since a positron's charge is positive, when the atom releases a positron it loses its positive charge and turns into a neutron.

When an atom emits a  (electron) particle, one of its neutrons releases a  (electron). And since an electron's charge is negative (-) the neutron loses its negative charge (since neutron's electrical charge is neutral) and turns into a proton.

This is simply explained ^ 

There are some other details but this should be enough :)