the universal cover
Among all the covers stacked over a space, there is a biggest one — the one that has unwound every loop. The circle's covers are the spiral ramps with finitely many floors and one infinite ramp; that infinite ramp, the real line, has no loops left at all. The universal cover is this maximal unwinding: the cover whose total space is simply connected, so that pi_1 has been completely dissolved.
Precisely, a universal cover of X is a covering p: X~ -> X with X~ simply connected (pi_1(X~) = 0). It is 'universal' because it covers every other connected cover of X: given any cover q: Y -> X, there is a covering map X~ -> Y making the triangle commute, so X~ sits at the top of the tower of all covers. By the Galois correspondence it corresponds to the trivial subgroup of pi_1(X, x_0), and its fiber over the basepoint is in bijection with pi_1 itself — the group acts on the fiber, and indeed the universal cover carries a free action of pi_1(X) by deck transformations with quotient X. Existence requires X to be path-connected, locally path-connected, and semilocally simply connected; under those hypotheses it exists and is unique up to isomorphism of covers.
The universal cover is the engine for computing and using pi_1. It realizes the fundamental group as a symmetry group (X = X~ / pi_1), it is the natural home for lifting any map out of a simply connected source, and it converts homotopy questions downstairs into rigid questions upstairs. Examples: R covers S^1 (pi_1 = Z); R^2 covers the torus (pi_1 = Z^2); the 2-sphere S^2 covers the projective plane RP^2 as a double cover (pi_1 = Z/2); the hyperbolic plane covers every closed surface of genus >= 2. An honest caveat: the universal cover need not be compact even when X is, and 'universal' is about covering all OTHER covers, not about being the largest space in any size sense — it is determined by simple-connectivity, full stop.
The universal cover of the torus T = S^1 x S^1 is the plane R^2, with covering map p(x, y) = (x mod 1, y mod 1). The deck group is Z^2 acting by integer translations, T = R^2 / Z^2, and indeed pi_1(T) = Z^2 — the deck group reads off the fundamental group directly.
R^2 unwinds the torus; the integer-translation deck group equals pi_1 = Z^2.
Existence is conditional: a space must be semilocally simply connected to have a universal cover. The Hawaiian earring (shrinking circles accumulating at a point) fails this and genuinely has no universal cover, so the slogan 'every space has one' is false.