An alpha particle is produced by the alpha decay of a radioactive nucleus. Because the nucleus is unstable a piece of it is ejected, allowing the nucleus to reach a more stable state.
The piece that is ejected is the alpha particle, which is made up of two protons and two neutrons: this is the nucleus of the helium atom. Helium is an inert and harmless gas, so the particles are not dangerous in themselves, it is only because of the high speeds at which they are ejected from the nuclei that they become dangerous. At these high speeds they have enough energy to break bonds in matter or ionise atoms (knock electrons off), which is especially deleterious for living cells.
An interesting thing about alpha particles is that they do not penetrate far through matter (unlike beta or gamma radiation). They are stopped by paper! Therefore an external source of alpha particles (or alpha radiation as it is called, for historical reasons, even though we now know they are particles) is not a big concern for humans.
Of course the alpha particles, while not penetrating far, deposit their considerable energy in a short distance (i.e. the piece of paper that stopped them absorbs all the energy). Hence, while our epidermis, the outer layer of our skin, is basically dead cells and so is not damaged by alpha particles, ingesting an alpha emitter, thereby giving it access to our internal organs would be a concern.
In fusion, helium/alpha particles are produced by the fusion reaction, along with neutrons. At JET we use magnetic confinement to contain the fusion reaction – because the helium nuclei are charged, they are confined by the magnetic field and do not escape. At the end of the experiment, when the magnets are turned off, these helium nuclei collide with the walls (which are thicker than paper!), recombine with some electrons, and return to being ordinary helium gas.
Note that the fusion reaction only continues to happen while the experiment is turned on – if the experiment gets too hot or cold, or gets too little fuel or too much then it stops. It is not like a spontaneously radioactive element (such as Americium-241 or Polonium-210).