The Java Stack class implements a last-in, first-out (LIFO) data structure called a stack. It extends the Vector class and inherits its methods. Elements are added to the top of the stack using push() and removed from the top with pop(). The search() method returns the position of an element from the top, and empty() checks if the stack is empty. Common uses of stacks in Java include storing and retrieving elements in LIFO order.