The eigenvector matrix C
$\mathbf{c}_n$ is a vector with $N$ entries (I switched to boldface to indicate - hopefully slightly more clearly - that it is a vector):

$$\mathbf{c}_n = \begin{pmatrix}c_{1n} \\ c_{2n} \\ \vdots \\ c_{Nn}\end{pmatrix}$$

This effectively means the same thing as writing

$$\psi_n = c_{1n}\phi_1 + c_{2n}\phi_2 + \cdots + c_{Nn}\phi_N = \sum_j c_{jn}\phi_j$$

I'll try to contextualise it. The columns of the eigenvector matrix are simply the eigenvectors themselves, so if (for example) the eigenvectors of an operator $\mathbf{A}$ are

$$\mathbf{c}_1 = \begin{pmatrix}p \\ q\end{pmatrix}; \qquad \mathbf{c}_2 = \begin{pmatrix}r \\ s\end{pmatrix}$$

then the corresponding eigenvector matrix $\mathbf{C}$ is 

$$\mathbf{C} = \begin{pmatrix} \mathbf{c}_1 &amp; \mathbf{c}_2 \end{pmatrix} = \begin{pmatrix} p &amp; r \\ q &amp; s \end{pmatrix}$$

[As an addendum: If the eigenvectors $\mathbf{c}_i$ are normalised, then you will find that

$$\mathbf{C}^{-1}\mathbf{A}\mathbf{C} = \begin{pmatrix} \lambda_1 &amp; 0 \\ 0 &amp; \lambda_2 \end{pmatrix}$$

where $\lambda_i$ is the eigenvalue of $\mathbf{c}_i$, i.e. $\mathbf{Ac}_i = \lambda_i\mathbf{c}_i$.]