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What does Laplace operator represent?

I was wondering what's the physical meaning of Laplace operator. $\vec\nabla$ actually represent a field. I had seen that Laplace operator is written as $\vec \nabla \cdot \vec\nabla=\nabla^2=\Delta$ So Laplace operator is scalar quantity.  I had found usage of Laplace operator in a book and [Wiki-page](https://en.wikipedia.org/wiki/Laplace_operator). 

I think the laplace operator is also representing "field" but it works like Wave. In a book I had seen that $$\nabla^2\vec A=\frac{1}{v^2}\frac{\partial^2 \vec A}{\partial t^2}$$

The above equation is representing wave. So does that mean Laplace operator represents wave?