The bra-ket notation generally consists of 'ket', i.e. a vector, and a 'bra', i.e. some linear map that maps a vector to a number in the complex plane.
Now, using this bra-ket notation we can compute the inner product of some operator, say $\hat{H}$, so $\langle\psi|\hat{H}|\psi\rangle$ defines the eigenvalue of some hermitian operator $\hat{H}$. This is also called the expectation value of $\hat{H}$ and describes the probability of measuring this operator given the state $\psi$. I hope this is correctly understood.
We can also derive the inner product $\langle\phi|\psi\rangle$. I must admit that I am a little confused about this representation although it makes sence mathematically. Does this mean the probability of being in the state $\phi$ given the state $\psi$? I hope someone can clarify.
Best Answer
The inner product is a thing between two vectors - "the inner product of some operator" is not a meaningful phrase. If $|\psi\rangle$ is a normalized eigenvector of $\hat H$ with eigenvalue $\lambda$, then it's true that $\langle \psi|\hat H|\psi\rangle = \lambda$, but the definition of an eigenvector/eigenvalue pair is that $\hat H|\psi\rangle = \lambda|\psi\rangle$.
$\langle \psi|\hat H|\psi\rangle$ is referred to as the expectation value (or expected value) of $\hat H$ (corresponding to the normalized state vector $|\psi\rangle$). The interpretation of this number is that if you take a large number of identical systems all prepared in the state $|\psi\rangle$ and measured $\hat H$ in each of them, you would expect the mean value of all of those results to be $\langle \psi|\hat H|\psi\rangle$.
There is no immediate physical interpretation of the inner product between two vectors - it is a quantity which shows up in all kinds of different contexts, and essentially measures the "overlap" between $\psi$ and $\phi$. It is analogous to the ordinary dot product between vectors in $\mathbb R^3$.
If $\psi$ is a normalized state vector representing the state of the system and $\phi$ is a normalized eigenvector of some observable $\hat A$ with (non-degenerate) eigenvalue $\lambda$, then $|\langle \phi|\psi\rangle|^2$ is the probability of measuring $\hat A$ to take the value $\lambda$. So that is one context in which the expression could arise. But trying to assign a single physical meaning to the inner product is like trying to assign a single physical meaning to the dot product between vectors in $\mathbb R^3$.