Your equation is the solution to Schrodinger's equation that describes a particle in a 1-D "box" of length L. The particular solution you have written represents the odd states. The integer-valued parametre, n, labels the quantum levels at which the particle can be inside the box. In each of these quantum states, 1, 2, 3,...$i$,...$j,...n,...$ the particle has different amount of energy, which are given by the equation
$E_n=\frac{h^2}{2mL^2}n^2$ where $n=1,2,3,...,i,...j,...n,...$.
A quantum state corresponding to the energy level $j$ is sybolically indicated by $|j>$ and is given by your equation with replacing $n$ by $j$, as you have done for $n=1$ and $n=2$. The particular expression you have written, $<\psi_i|\psi_j>$, represents the transition probability amplitude for the particle to jump from quantum state $|\psi_j>$ to $|\psi_i>$. The latter is zero since these wave-functions form an orthonormal set of quantum states, and you are showing this with the $\delta_{ij}$ symbole.