path-connected
A space is path-connected if you can walk continuously from anywhere to anywhere without leaving it — between any two points there is a road, drawn inside the space, that you can trace without lifting your pen. It is the most hands-on form of connectedness: not merely “in one piece” but “navigable in one piece”.
Formally, X is path-connected if for every pair of points p and q there is a continuous map gamma from the closed interval [0, 1] into X with gamma(0) = p and gamma(1) = q. Such a gamma is a path from p to q. Since the interval [0, 1] is connected and continuous images of connected sets are connected, path-connected always implies connected.
The converse fails, and the standard cautionary tale is the topologist's sine curve — the graph of sin(1/x) for x in (0, 1] together with the segment of the y-axis it accumulates on. That set is connected (you cannot separate the wildly oscillating graph from the limiting segment by open sets) yet not path-connected (no continuous path can cross from the oscillating part to the segment). For open subsets of R^n, however, the two notions coincide.
Any convex set in R^n — a ball, a cube, the whole plane — is path-connected: join p and q by the straight segment gamma(t) = (1 - t) p + t q, which stays inside the set and is obviously continuous. So convexity is a quick sufficient (not necessary) condition for path-connectedness.
In a convex set the straight segment between two points is a path, so convex sets are path-connected.