As we get on in quantum theory, the need for the long integral equations starts to disappear when we recognize that so much can be done solely by knowing that eigenstates are orthogonal and by asserting that they have been normalized. Furthermore, when we know the eigenvalues, those labels carry all the information we need to know. Because of this, Dirac developed a simplified notation. It is also called the bracket notation. The symbol |n> is called a ket. It is the same as the wavefunction yn. The symbol <n| is called a bra. It is the same as the complex conjugate yn*. If we use the operator A to operate on |n>, and then left multiply by <n| and integrate, we have <n|A|n> which is the expectation value equation we ran into before. This last entity is called a bracket (do you see the bra - c - ket?) If the operator is 1,then the operator is just left out to give <m|n>, for instance.
To see Dirac notation in operation, take a look at the first two proofs of the hermitian operator properties given previously in Operator Notation.