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Beta particles are electrons (symbol β-), or positrons (symbol β+), emitted in beta decay (a kind of radioactivity); beta radiation in other words. Sometimes ‘beta particles’ refers to high energy electrons, irrespective of their source (e.g. the beta particles in the Van Allen radiation belts around the Earth; very few are produced by beta decay).
Of the three kinds of radioactivity (alpha, beta – both of which are particles – and gamma (which is electromagnetic radiation)), beta particles have intermediate penetrating power.
Beta particles have an important role in medicine … as diagnostic tools, to treat some diseases (notably various cancers, particularly via radionuclide therapy), in biochemical analysis, etc. For example, 18F (the fluorine-18 isotope) is used as a positron (β+) emitter in positron emission tomography (PET).
Beta particles – or rather the weak interaction which is the cause of their emission – were crucial in Big Bang Nucleosynthesis … as the early universe cooled, reactions between the protons, neutrons, electrons, and photons produced many light nuclides, but the balance between many reactions left only hydrogen, deuterium, helium-3, helium-4, and lithium-7 when the universe became too cool for any nuclear reactions to continue (of course, isolated neutrons and unstable nuclides – such as tritium – were also left, but they decayed well before the time of cosmic microwave background).
Fast forward to today … beta (β+) particles from the decay of potassium-40 is one source of internal heat for the Earth – giving us plate tectonics, its magnetic field, etc – … the decay of carbon-14 and beryllium-10 (both of which produce beta (β-) particles) provide us with tools to do radioactive (or radiometric) dating (these are two of the nuclides produced by cosmic ray spallation).
Beta Particle Radiation is a good, introductory webpage (from the University of California, Davis), and Weak Interactions explains how beta particles and the weak (nuclear) force are related (from SLAC’s Virtual Visitor Center)
Universe Today has several stories which cover the role of beta particles in astronomy; for example A Prototype Detector for Dark Matter in the Milky Way, and Fermilab Putting the Squeeze on Higgs Boson.