The β (beta) of a security is essentially a measure of the strength of the relationship between the price of a security and the market. It is used to calculate discount rates for CAPM.
A high beta security will tend to move more than the market, and a low beta security less. This is sometimes presented as a definition of beta. This correct to an extent, but is so vague as to be misleading.
The beta is defined using two relatively straightforward statistical measures.
where covs,m is the covariance of the return of the security with the market return, and,
varm is the variance of the market return.
Market, strictly speaking, means a market portfolio: all of every available security. In practice an index is used rather than calculating this directly.
Beta may be calculated for a portfolio (portfolio beta) rather than for a single security, usually for performance measurement purposes, another key use of beta.