Data structures organize data in a specialized format for efficient organization, processing, storage, and retrieval of information. They include primitive data types like integers, floats, characters, and booleans that occupy fixed memory sizes, as well as non-primitive types like arrays that can store sequential collections of elements of the same type. Common data structures are arrays, sparse matrices with mostly zero elements, and stacks which follow last-in first-out ordering for operations like push to insert and pop to remove top elements.