There are two main kinds of nuclear reactions: fission and fusion.
In fission, a heavy atomic element is split and some of the mass is given off as energy. Not that it is not "destroyed", since mass and energy are always conserved. The missing mass is converted to energy.
Take as an example the U-235 isotope of uranium. A fission reaction starts when it is struck by a neutron. The atom splits, releasing three neutrons in the process and some energy. The energy released is about 0.1% of the mass of the nucleus, and much of the missing mass is released as photons.
In fusion, the atoms of hydrogen are fused together to form helium with a release of energy. In one kind of reaction, two deuterium atoms combine to form helium with a release of a neutron and a relatively large amount of energy as photons. In stellar fusion, the process that powers stars like our Sun, hydrogen nuclei as protons (without neutrons) are fused to form helium. This process releases a number of energetic particles, including neutrinos, positrons, and gamma rays (highly energetic photons). (Notice that the positron has a positive charge. I order to conserve charge, some of the protons in stellar fusion become neutrons, losing their positive charge. )
There is one more type of nuclear reaction: radioactive decay. The particles released here depend on the type of element decaying. Some typical release particles might include alpha particles (2 neutrons plus 2 protons), beta particles (electrons), and gamma rays.