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So an action G on X is (G,X) -> X. so here G would be pi_1(X, x0) and X would be some point in p^-1(x0) and this action gives another point in p^-1(x0). The deck transformation is defined to be a homeomorphism of the covering space with itself (where universal cover of S1 x S1 is R x R here). So I can see what both are doing geometrically but what does it for the actions to be the same really? how do i tackle this.

I'm also not sure about abelian, I think it means f*g and g*f have to be homotopic where f,g, are two loops in pi_1(X, x0) where is this useful usually.