A werecreature is classified as a beta simply by being an accepted member of a pack. They maintain a mystical symbiotic bond with their packmates, thus becoming more powerful and mature as were-creatures. Though the rules of pack membership in the show appear flexible and are ill-defined, a werecreature could count as a beta simply by mutual and accepting association with an alpha or another experienced beta.
Betas can be born to werewolf parents. Derek Hale, and his sisters, Laura and Cora all inherited the werewolf trait from their mother Talia.
They can also be formerly humans who have been bitten by an alpha. In Season 1, Scott McCall became a beta werewolf right after being bitten by Peter Hale. Out of necessity and convenience, Scott and Derek formed a tenuous alliance to stop Peter and so they were classified as betas.
Like Alphas, they are able to fully transform their bodies further into more anthropomorphic shapes and such as a full wolf as in the case of Derek Hale. However, this is harder, more challenging for them to accomplish because they lack the spark of power an alpha possesses that supplements the ability to do so.
The ill-defined rules of pack membership not withstanding, if a beta were to dissassociate, cut themselves off from a pack, whether by choice or as punishment, or conspiring events, their ability, power and control would accordingly diminish. They would then be classified as omegas.