How are two matrices similar?

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I am aware that this question has been asked before, but I would like to see if anyone can come up with a better explanation.

So what I have been stuck on is, what are the requirements for two matrices to be similar.

I know that Matrix A and B are similar if the following is true: $A = V^{-1} \cdot B \cdot V$ But how in the world do I find V.

It is one thing to determain V, but how do I know that such a vector exists. In other words, what is the requirements for two matrices to be similar.