An equivalent definition is to require, for all x and y in an algebra A, that x(xy) = (xx)y and (xy)y = x(yy). The equivalence of the two definitions is known as Artin's Theorem.
For any two elements x and y in an alternative algebra another simple identity holds: (xy)x = x(yx).
Every associative algebra is obviously alternative, but so too are some non-associative algebras such as the octonions.
Alternativity is a condition in-between associativity and power associativity.