the fundamental group
The fundamental group is topology's way of detecting holes by looking at loops. Pick a base point on your space and consider all the loops that start and end there — round trips you can take, like a piece of string pinned at one end and led out and back. On a flat disk every loop can be slowly reeled in and shrunk down to the base point; the space has no holes to catch the string. But on a doughnut, a loop that goes around the hole cannot be shrunk away without leaving the surface — the hole holds it. The fundamental group catalogues exactly which loops are essentially different and which can be deformed into one another.
Made precise: fix a base point p, and call two loops at p equivalent if one can be continuously deformed (homotoped) into the other while keeping the base point pinned. Each equivalence class is a 'kind of loop'. You can combine two loops by doing one and then the other — first travel loop a, then loop b — and this combination turns the set of loop-classes into a group, written pi_1(X, p): there is an identity (the loop that just sits still), every loop has an inverse (run it backwards), and the operation is associative. The structure of this group encodes the space's holes. For the circle, pi_1 is the integers Z, the integer counting how many times (and which way) a loop winds around; for the plane or a disk, pi_1 is trivial — just the identity.
The fundamental group is the first great algebraic invariant of topology: it turns the soft, geometric question 'are these spaces the same?' into the hard, computable question 'are these groups the same?'. If two spaces are homeomorphic their fundamental groups match, so different groups prove different spaces. A space whose fundamental group is trivial — every loop shrinks to a point — is called simply connected; the sphere and the plane are simply connected, the circle and torus are not. The honest caveat: the fundamental group is a complete answer for loops but does not see every kind of hole. The 2-sphere has trivial pi_1 (every loop on it shrinks) yet is clearly not the same as a point — its 2-dimensional 'cavity' is detected only by higher invariants, not by pi_1.
On a circle, label loops by their winding number: the loop that goes around once counterclockwise is 1, twice is 2, once clockwise is -1, and standing still is 0. Composing loops just adds the winding numbers (going around twice then once more makes three times: 2 + 1 = 3), so pi_1(circle) is exactly the integers Z under addition. On a flat disk, by contrast, every loop reels in to a point, so its winding has nowhere to register and pi_1 is trivial — the algebra has detected that the disk has no hole and the circle has one.
pi_1 of the circle is the integers (winding number); pi_1 of the disk is trivial — algebra counting holes.
The fundamental group sees only loops, so it can miss higher-dimensional holes: the 2-sphere has trivial pi_1 yet is not a point, since its hollow interior is detected by higher homotopy or homology groups, not by pi_1.