the fundamental group
Stand at a fixed spot in a space and walk out along a loop that returns to where you started. In a flat field you can always reel any such loop back in to a point. But on the surface of a donut, a loop that goes around the hole cannot be shrunk away no matter how you tug it. The fundamental group is the bookkeeping device that records exactly which loops can be deformed into which others, and it is the first and most basic algebraic fingerprint of a space's shape.
Fix a basepoint x_0 in X. Consider loops based at x_0: continuous maps gamma: [0,1] -> X with gamma(0) = gamma(1) = x_0. Two loops are identified when they are homotopic rel endpoints (deformable into one another while keeping both ends pinned at x_0). The set of these homotopy classes is pi_1(X, x_0). It becomes a group: the product [alpha][beta] is the class of 'walk alpha then walk beta' (concatenation, traversed at double speed), the identity is the constant loop at x_0, and the inverse of [gamma] is [gamma run backwards]. Associativity and the inverse law hold only up to homotopy — and that is precisely why we pass to homotopy classes, where the wiggle room makes them hold exactly.
The fundamental group is functorial: a based map f: (X, x_0) -> (Y, y_0) induces a homomorphism f_*: pi_1(X, x_0) -> pi_1(Y, y_0) by [gamma] -> [f ∘ gamma], with (g ∘ f)_* = g_* ∘ f_* and (id)_* = id. Hence homotopy-equivalent spaces have isomorphic fundamental groups, making pi_1 a genuine invariant. Changing the basepoint to x_1 (in the same path-component) gives an isomorphism pi_1(X, x_0) ≅ pi_1(X, x_1), but the isomorphism depends on a chosen path between the basepoints and is generally only well-defined up to inner automorphism — so for non-abelian pi_1 the group is canonical but the identification across basepoints is not. A space with pi_1 trivial is called simply connected; pi_1 is the precise obstruction to simple connectivity.
For the figure-eight (wedge of two circles), pi_1 is the free group on two generators a and b: a loop is recorded by the sequence of left- and right-circle traversals, like a b a^{-1} b, and no nontrivial reduced word is null-homotopic. This non-abelian pi_1 already distinguishes the figure-eight from the torus, whose pi_1 = Z x Z is abelian.
The figure-eight has a free, non-abelian fundamental group — order of traversal matters, unlike on the torus.
pi_1 generally depends on the path-component containing the basepoint and need not be abelian; only the higher groups pi_n for n >= 2 are automatically abelian. Quoting 'pi_1(X)' without naming a basepoint is harmless only when X is path-connected.