What nuclear process occurs that marks the official birth of the star?

Stars are formed from large clouds of gas and dust which collapse under gravity. When this happens, the core starts rotating and attains high temperatures. When the temperatures at the core become high enough, a nuclear process takes place in which lighter nuclei combine to form heavier nuclei. This is the process of nuclear fusion and typically, hydrogen atoms fuse together to form helium atoms. Once the nuclear fusion process starts, this collapsing mass of gases and dust starts generating energy which pushes outwards. The two forces--inward gravity of the core and outward motion of energy--balance each other. Over time the energy generated from the nuclear fusion becomes high enough to push the outer envelope of gases away and the star starts giving off light energy. 

The other form of nuclear reaction is nuclear fission, which we commonly use in nuclear reactors to produce electricity. In this process, heavier nuclei are broken into lighter nuclei.

Hope this helps.