This is a very broad question, and you start in the wrong end. The different variance-covariance matrices is implied by your model assumptions, so you should start there.
Lets say you have many short time series (maybe different subject followed-up over some short time). We assume this time series are stationary (time trends, if they exist, taken care of by other parts of the model). If we in addition assume this time series are autoregressive of some order, then the variance-covariance matrix will have constant values on sub/super-diagonals parallel with the main diagonal. Such matrix is called a Toeplitz matrix https://en.wikipedia.org/wiki/Toeplitz_matrix
One other example: If you have a model with random intercepts for (many) small groups, then the covariance matrix will have only two different values, one on the main diagonal and another (smaller) outside the main diagonal. IThat is called an exchangeable covariance matrix. I leave out the calculations.
So: start making your conceptual model, and then the (form of) covariance matrix will follow.