a contractible space
A contractible space is one that can be shrunk continuously, within itself, down to a single point. Picture a lump of clay you can slowly collapse onto one of its own grains without ever tearing it; a solid ball does this, but a coffee-cup-with-handle (a torus) does not, because the hole gets in the way. Contractibility is the homotopy-theoretic way of saying a space 'has no holes of any kind' — it is as simple as a point can make it.
Formally, X is contractible if the identity map id_X is homotopic to a constant map. Equivalently, X is homotopy equivalent to a one-point space {*}: there is a map to the point and a map back (picking any basepoint) whose composites are each homotopic to the identity. The homotopy H: X x [0,1] -> X with H(x, 0) = x and H(x, 1) = x_0 for some fixed x_0 is called a contraction; reading it the other way, every map into a contractible space, and every map out of one onto a 'nice' target, behaves as trivially as possible up to homotopy.
Contractible spaces are the trivial objects of the theory: all their homotopy and (reduced) homology groups vanish, pi_n = 0 for every n, and any two maps into a contractible space are homotopic. Every convex (indeed every star-shaped) subset of R^n is contractible by the straight-line homotopy to a centre. Two honest cautions: contractible is NOT the same as simply connected — simply connected only kills pi_1, while contractible kills every pi_n — and a contractible space need not be a point or even look simple (the infinite-dimensional sphere S^infinity is contractible, and there are contractible 2-complexes that are far from obviously so).
Any convex set C in R^n — a disk, a cube, a half-space, all of R^n itself — is contractible: fix a point c_0 in C and use H(x, t) = (1 - t) x + t c_0, which stays inside C by convexity and shrinks everything to c_0 at t = 1.
Convexity gives contractibility for free; this is why so many local pictures in geometry (charts, stars of vertices) are homotopically trivial.
Do not equate contractible with simply connected: simple connectivity is only pi_1 = 0, whereas contractibility forces every pi_n = 0; S^2 is simply connected but very much not contractible (pi_2(S^2) = Z).