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.
β= (covs,m)/(varm)
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.