Non-Commutative Algebra is a branch of mathematics that studies algebraic structures where the order of operations matters. In contrast to commutative algebra, where the equation a \cdot b = b \cdot a holds for all elements a and b, non-commutative algebra involves structures like matrices and quaternions, where changing the order can lead to different results. This area of algebra has applications in various fields, including quantum mechanics and theory of groups. Non-commutative structures are essential for understanding complex systems and phenomena, making them a vital part of modern mathematical research and applications.