Intuitively, a continuous function is one where small changes of input result in correspondingly small changes of output. Use this tag for questions involving this concept. As there are many mathematical formalizations of continuity, please also use an appropriate subject tag such as (real-analysis) or (general-topology)
Analytic Definition: Let $(X, d_X)$ and $(Y, d_Y)$ be metric spaces. A map $f : X \to Y$ is said to be continuous at $x_0$ if for every $\varepsilon > 0$, there is $\delta > 0$ such that $d_Y(f(x_0), f(x)) < \delta$ whenever $d_X(x_0, x) < \varepsilon$. A map $f : X \to Y$ is said to be continuous if it is continuous at $x_0$ for all $x_0 \in X$.
Topological Definition: Let $(X,\mathcal T_X)$ and $(Y, \mathcal T_Y)$ be topological spaces. A map $f : X \to Y$ is said to be continuous if $U \in \mathcal T_Y$ implies that $f^{-1}(U) \in \mathcal T_X$.
In the case of metric spaces, the metric induces a topology, and the two notions of continuity coincide. Note that multiple metrics can induce the same topology, and that not all topologies are metrizable (can be generated from some metric).
Continuity is a sufficient condition for the intermediate value theorem. It is also a necessary condition for the extreme value theorem, as well as differentiability.