Please, would someone be so kind and explain what exactly happens when Singular Value Decomposition is applied on a matrix? What are singular values, left singular, and right singular vectors? I know they are matrices of specific form, I know how to calculate it but I cannot understand their meaning.

I have recently been sort of catching up with Linear Algebra and matrix operations. I came across some techniques of matrix decomposition, particularly Singular Value Decomposition and I must admit I am having problem to understand the meaning of SVD.

I read a bit about eigenvalues and eigenvectors only because I was interested in PCA and I came across diagonalizing a covariance matrix which determines its eigenvectors and eigenvalues (to be variances) towards those eigenvectors. I finally understood it but SVD gives me really hard time.

thanks