What is the difference between "singular value" and "eigenvalue"? I am trying to prove some statements about singular value decomposition, but I am not sure what the difference between singular value and eigenvalue is.

Is "singular value" just another name for eigenvalue?
The singular values of a $M\times N$ matrix $X$ are the square roots of the eigenvalues of the $N\times N$ matrix $X^*\,X$ (where $^*$ stands for the transpose-conjugate matrix if it has complex coefficients, or the transpose if it has real coefficients). 

Thus, if $X$ is $N\times N$ real symmetric matrix with non-negative eigenvalues, then eigenvalues and singular values coincide, but it is not generally the case!
The eigenvalues $\mathbf{\Lambda}$ of the sample matrix $\mathbf{X}$ can be obtained applying SVD to the covariance of $\mathbf{X}$:

$$\mathbf{S} = cov(\mathbf{X}) \mapsto \mathbf{S}=\mathbf{U}\mathbf{\Lambda}\mathbf{V}^{T} $$