If a two-electron covalent bond breaks symmetrically, each of the two atoms receive one electron; it is a homolytic bond cleavage.
A homolytic cleavage is shown using a fish-hook arrow, which implies one-electron movement.

The atom possessing the odd or the unpaired electron is called a free radical, denoted with a dot over the element (E.). The free radicals are unstable, reactive, electron-deficient species. They are incapable of independent existence and are, therefore, short-lived.
Homolytic bond cleavage is seen in non-polar bonds under conditions of high temperatures, Ultraviolet radiations, and electrolysis and requires radical initiators like peroxides.
Importance of free radicals
Some of the halogenation (replacement of hydrogen of alkane with halogen) reactions in chemistry proceed by homolytic cleavage where the free radicals are intermediates. Examples include alkyl halogenations, benzylic and allylic halogenation, and Anti-Markovnikov reactions.

The reaction always occurs in three steps- initiation, propagation, and termination.
Free radicals have beneficial and harmful effects on our bodies, regulated by antioxidants. They are produced in the body as part of cell metabolism and play a role in cell growth and killing harmful pathogens. In excess, they are detrimental and linked to many cancers, autoimmune diseases, aging, and cardiovascular and neurodegenerative conditions.
This excerpt is additional content inspired by the chapters Reactive Intermediates, Covalent Bond and Bonding in Atoms, which are part of CurlyArrows' Introductory Organic Chemistry Course. Preview the Book.
Related Reading - Heterolytic Bond Cleavage.