A groupoid is connected if it is inhabited and every object is connected by a morphism to every other object.
Every category $C$ induces a groupoid $G(C)$ by freely inverting all its morphisms. A category is connected if the groupoid $G(C)$ is.
A category $C$ is connected if it is inhabited and the following equivalent conditions hold
the $\infty$-groupoidification of the category (the Kan fibrant replacement of its nerve) is a connected ∞-groupoid.
the geometric realization of its nerve is a connected topological space.
the localization $C[C_1^{-1}]$ of $C$ at all its morphisms is a connected groupoid.
Note that the empty category is not connected. For other purposes, one can argue about whether the empty set should be called “connected” (see connected space), but for the applications of connected categories, the empty category should definitely not be called connected. In particular, a terminal object is not a connected limit.
A connected limit is a limit whose domain diagram category is connected.