Understanding gauge transformations and relation to Lie Groups?

Question

I am starting to study gauge theory. I have a background in basic group theory, multivariable calculus, and the idea of symmetry in relation to group theory. I am trying to understand why the electromagnetic potential 4-vector must undergo the transformation $\mathbf{A}\mapsto\mathbf{A}-\nabla\alpha$. In trying to understand where the $\nabla\alpha$ term came from, I googled gauge transformations to understand acceptable transformations, and Wikipedia stated that a gauge transformation is a theory where the Lagrangian is invariant under Lie Group operation transformations. What exactly classifies something as a Lie Group? What symmetry(ies) does it represent (my understanding is that every group is "caused" i.e. representative of some symmetry)?