Let p \colon \tilde{X} \to X be a covering map with \tilde{X} path-connected and X path-connected and locally path-connected. A deck transformation is a homeomorphism h \colon \tilde{X} \to \tilde{X} with p \circ h = p, so it permutes each fiber and carries sheets to sheets. The deck transformations form a group G(\tilde{X}) under composition [1,2].
A deck transformation is a lift of p itself, so by the uniqueness of lifts it is determined by the image of a single point. In particular a deck transformation with a fixed point is the identity, and G(\tilde{X}) acts freely on each fiber. For \mathbb{R} \to S^1 the deck transformations are the translations s \mapsto s + k with k \in \mathbb{Z}, for z \mapsto z^n the rotations by multiples of 2\pi/n, and for S^n \to \mathbb{RP}^n the identity and the antipodal map.
Moving the base point within the fiber changes H = p_*\pi_1(\tilde{X}, \tilde{x}_0) by conjugation: if a path from \tilde{x}_0 to \tilde{x}_1 projects to a loop \gamma, then p_*\pi_1(\tilde{X}, \tilde{x}_1) = [\gamma]^{-1} H [\gamma]. By the lifting criterion applied to p itself, a deck transformation sending \tilde{x}_0 to \tilde{x}_1 exists exactly when these two subgroups are equal [1].
The covering is normal if for any two points of a fiber some deck transformation sends one to the other, that is, if G(\tilde{X}) acts transitively on each fiber. This happens exactly when H equals all of its conjugates, that is, when H is a normal subgroup of \pi_1(X, x_0), and then G(\tilde{X}) \cong \pi_1(X, x_0)/H. In general G(\tilde{X}) \cong N(H)/H, where N(H) is the normalizer of H [1]. For the universal cover, H is trivial and G(\tilde{X}) \cong \pi_1(X, x_0).
The figure compares two 3-sheeted coverings of the figure eight S^1 \vee S^1, drawn as graphs whose vertices 1, 2, 3 lie over the base point. In the first, both a and b move 1 to 2, 2 to 3 and 3 to 1, and turning the graph by a third of a full turn carries every edge onto an edge with the same label and direction. These rotations are the deck transformations, G(\tilde{X}) \cong \mathbb{Z}/3\mathbb{Z} acts transitively on the vertices, and the covering is normal.
In the second, b exchanges 1 and 2 and fixes 3. The same rotation no longer matches the edges labeled b, and no symmetry of the graph other than the identity respects the labels and directions, so G(\tilde{X}) is trivial. The subgroup H consists of the words whose lift from 1 ends at 1. It contains a^{-1}ba, whose lift runs from 1 to 3, stays at 3 and returns to 1, but not its conjugate b, whose lift ends at 2, so H is not normal.
Assume now that X is path-connected, locally path-connected and semilocally simply connected, so that it has a universal cover \tilde{X} \to X with deck group \pi_1(X, x_0). For each subgroup H \subset \pi_1(X, x_0), the quotient \tilde{X}/H of the universal cover by the deck transformations in H is a covering space of X, and its subgroup is exactly H [1]. In terms of path classes, it identifies [\gamma] with [h\gamma] for h \in H.
This is the Galois correspondence. Sending a covering to p_*\pi_1(\tilde{X}, \tilde{x}_0) is a bijection between connected coverings with a chosen point over x_0, up to isomorphisms that keep that point, and subgroups of \pi_1(X, x_0) [1]. If the chosen point is forgotten, coverings correspond to conjugacy classes of subgroups, and normal coverings correspond to normal subgroups.
The correspondence reverses size. The trivial subgroup gives the universal cover, and the whole group gives X itself with one sheet. If H \subset K, the covering for H factors through the covering for K by the lifting criterion, and the number of sheets of the first over the second is the index of H in K. This is the pattern of Galois theory, where a smaller subgroup of the Galois group fixes a larger field.
For the torus S^1 \times S^1, whose universal cover is \mathbb{R}^2 with deck group \mathbb{Z}^2, every subgroup is normal. A subgroup of finite index n gives a torus with n sheets, a subgroup isomorphic to \mathbb{Z} gives a cylinder, and the trivial subgroup gives \mathbb{R}^2. The figure colors the unit squares of \mathbb{R}^2 by the cosets of H in \mathbb{Z}^2: the squares of one color become one sheet, and one square of each color makes up the covering torus.
The subgroups 2\mathbb{Z} \times \mathbb{Z} and \langle (1, 1), (1, -1) \rangle have index 2 and give vertical stripes and a checkerboard. Both contain 2\mathbb{Z} \times 2\mathbb{Z}, of index 4, whose four colors refine both patterns. So the 4-sheeted torus is a 2-sheeted covering of each of the two 2-sheeted tori.
For the figure eight, \pi_1 is the free group F_2, and the correspondence is much richer. The two coverings of the previous figure come from subgroups of index 3. The first is normal, while the second corresponds to three different conjugate subgroups, one for each vertex chosen as the base point.
Coverings can also be built from group actions. Suppose a group G acts on a space Y by homeomorphisms so that every point has a neighborhood U such that gU and U are disjoint for all g \neq 1, a covering space action. Then the quotient map Y \to Y/G is a normal covering, and if Y is path-connected, its deck group is exactly G [1]. If Y is moreover simply connected and locally path-connected, Y \to Y/G is the universal cover and \pi_1(Y/G) \cong G.
Translations by \mathbb{Z}^2 on \mathbb{R}^2 give the torus, and the antipodal map of S^n gives \mathbb{RP}^n, so \pi_1(\mathbb{RP}^n) \cong \mathbb{Z}/2\mathbb{Z} for n \geq 2. On S^3 \subset \mathbb{C}^2, the map (z, w) \mapsto (\omega z, \omega^q w) with \omega = e^{2\pi i/p} and q coprime to p generates a free action of \mathbb{Z}/p\mathbb{Z}, whose quotient, the lens space L(p, q), has fundamental group \mathbb{Z}/p\mathbb{Z}.
The condition on neighborhoods cannot be weakened to the absence of fixed points. The rotation of the circle by an irrational multiple of 2\pi generates a free action of \mathbb{Z}, but every orbit is dense, so no neighborhood is disjoint from its images, and the quotient is not even a Hausdorff space.
The figure shows a free action of \mathbb{Z}/2\mathbb{Z} on the torus. With the torus placed symmetrically around the origin of \mathbb{R}^3, the map x \mapsto -x sends it to itself, turning each point half a turn around the axis and reflecting its angle around the tube. The origin lies in the hole, so no point is fixed, and a finite group acting freely on a Hausdorff space always gives a covering space action.
As the point P goes once around a circle of latitude, -P goes around the latitude at the opposite height, and as P goes once around the tube, -P goes around the tube on the far side in the opposite direction. The map reverses orientation, and the quotient is the Klein bottle K. The torus is therefore a 2-sheeted normal covering of K, and \pi_1(T^2) \cong \mathbb{Z}^2 is a normal subgroup of index 2 in \pi_1(K).