Bra-ket notation is concise and useful.

A wavefunction is represented by a ket $|\psi\rangle$.

The complex conjugate of wave function is written as a bra $\langle\psi|$.

The complex conjugate of a variable is found by swapping the sign of the imaginary part of said variable’s complex number, in other words: reflecting z across the real axis. For example,

$$z = x + iy$$

$$z^* = x – iy$$

A bra on the left and a ket on the right implies integration over dt.

$$\langle\psi|\psi\rangle \equiv \int\psi^*\psi dt$$

$$\langle\psi|\psi\rangle \equiv \int\psi^*\psi dt$$

Similarly

$$\langle\psi|\hat{X}|\psi\rangle \equiv \int\psi^*\hat{X}\psi dt$$

My brief tutorial covered the basic usages of bra-ket notation in a quantum mechanical context; bra-ket notation is also used elsewhere.