When an atom of beryllium-9 is bombarded by an alpha particle, an atom of carbon-12 is produced and a neutron is emitted. what kind of nuclear reaction just occurred?

Nuclear fusion

Explanation:

By definition a nuclear fusion is a reaction in which two or more atomic nuclei are combined to form one or more different atomic nuclei and subatomic particles (neutrons or protons).  

In this case the two nuclei are the nucleus of atom of beryllium-9 (which has 4 protons and 5 neutrons) and the alpha particle (which has 2 protons and 2 neutrons), their combination produce the nucleus of carbon-12 (which has 6 protons and 6 neutrons) and a neutron.