A Banach algebra is a type of algebraic structure that combines the properties of a normed space and an algebra. It consists of a set of elements where you can perform addition, multiplication, and scalar multiplication, while also having a way to measure the size of elements through a norm. This norm must satisfy certain conditions, making the space complete, meaning that every Cauchy sequence converges within the space. In a Banach algebra, the multiplication operation is associative, and there is a multiplicative identity element. These algebras are important in functional analysis and have applications in various fields, including quantum mechanics and signal processing. They help in studying linear operators and their properties.