A topological space $X$ is path-connected (or pathwise connected) if, for any $a, b \in X$, there exists a path from $a$ to $b$. That is, if there exists a continuous mapping $f:[0,1]\rightarrow X$ such that $f(0)=a$ and $f(1)=b$.
This is closely related to arc-connected spaces, in which there is an arc between any two points. That is, for any $a, b \in X$, there is a path from $a$ to $b$ which is homeomorphic to the unit closed interval.