the fundamental group of the circle
The single most important calculation in beginning homotopy theory is that the loops on a circle are classified by an integer: how many times the loop winds around, counting direction. Wind once counterclockwise and you cannot undo it; wind once and then once back and the two cancel. This integer is the winding number, and the statement pi_1(S^1) ≅ Z says it is a complete invariant of a loop up to deformation.
The proof is the model example of the lifting machinery. Use the covering map p: R -> S^1, p(s) = (cos 2 pi s, sin 2 pi s), which wraps the real line around the circle, with the integers sitting above the basepoint 1. Any loop gamma in S^1 based at 1 lifts uniquely to a path gamma~ in R starting at 0 (this is the path-lifting property of a covering). Because gamma is a loop, its lift ends at some integer n = gamma~(1); this n is the winding number, and the map [gamma] -> n is a well-defined homomorphism pi_1(S^1) -> Z by the homotopy-lifting property. It is surjective (the loop s -> n s winds n times) and injective (a loop winding 0 times lifts to a loop in R, which is contractible, so the original is null-homotopic). Hence the map is an isomorphism.
This one fact powers a surprising amount of mathematics. It is the engine behind the Brouwer fixed-point theorem in dimension two, the fundamental theorem of algebra (a degree-n polynomial winds n times around large circles, so it must have a root), and the non-existence of a retraction of the disk onto its boundary. More structurally, it is the first nontrivial covering-space computation and the template for computing pi_1 of any space built from circles. An honest note: the slick 'wind n times' picture is correct, but the rigor lives entirely in the unique path lifting and homotopy lifting — without those the winding number is not even well-defined on homotopy classes.
The loop gamma(t) = (cos 4 pi t, sin 4 pi t) lifts to gamma~(t) = 2t in R, ending at 2, so [gamma] corresponds to 2 in Z — it winds twice. Concatenating it with a once-clockwise loop (lift ending at -1) gives a loop lifting to end at 1, hence class 1, exactly as 2 + (-1) = 1 in Z.
Winding numbers add under concatenation, matching addition in Z — the isomorphism pi_1(S^1) = Z made concrete.
The result needs the full covering R -> S^1; a finite cover like the n-fold self-cover S^1 -> S^1 detects winding only modulo nothing useful, and you cannot read off Z from a finite cover. The universal cover R, being contractible, is what forces injectivity.