a covering space
Think of a spiral parking ramp viewed from directly above: the shadow is a single circular drive, but the ramp itself is many floors stacked over it, and walking the circular drive once takes you up exactly one level. A covering space is the precise version of this 'space stacked evenly over another': it spreads out the loops of the base so that what tangles below becomes untangled above.
A covering map is a continuous surjection p: X~ -> X such that every point x in X has an open neighborhood U that is evenly covered: p^{-1}(U) is a disjoint union of open 'sheets,' each mapped homeomorphically onto U by p. The space X~ is the covering space (the total space), X is the base, and p^{-1}(x) is the fiber over x — a discrete set whose cardinality (constant on each component of a connected base) is the number of sheets. Two defining properties follow and do the real work: path lifting (every path in X starting at a point lifts uniquely to X~ once you choose a start in the fiber) and homotopy lifting (homotopies of paths lift too, compatibly). The example p: R -> S^1 wrapping the line onto the circle, with fiber Z, is the prototype.
Covering spaces are the geometric face of the fundamental group. The induced map p_*: pi_1(X~, x~_0) -> pi_1(X, x_0) is always injective, and its image is a subgroup of pi_1(X, x_0); the index of that subgroup equals the number of sheets. This sets up the Galois correspondence between connected covers of a reasonable space and subgroups of pi_1, with the universal cover (simply connected total space) sitting over the whole group. Honesty about hypotheses: the clean classification needs the base to be path-connected, locally path-connected, and semilocally simply connected — pathological spaces (like the Hawaiian earring) fail the last condition and have no universal cover in the usual sense.
The map p: S^1 -> S^1, p(z) = z^n (writing the circle as unit complex numbers), is an n-sheeted cover: each point has n preimages, the n-th roots of it. Here p_* sends a generator of pi_1(S^1) = Z to n times a generator, so the image is the subgroup nZ, of index n — matching the n sheets.
The degree-n self-cover of the circle realizes the subgroup nZ of pi_1 = Z; index equals number of sheets.
Not every surjective local homeomorphism is a covering: the map (0, 2) -> S^1 wrapping an open interval around fails to be evenly covered near the basepoint (the fiber there is not 'spread into sheets'). The even-covering condition over EVERY point is essential, not a formality.