If $X$ is a Hausdorff topological space and it is path-connected, then it is arcwise-connected.
A path-connected Hausdorff space is arc-connected. I don't know (but would like to) any simple proofs of this claim. One way is to prove that every Peano (meaning compact, connected, locally connected and metrizable) space is arc-connected and then note that the image of a path in a Hausdorff space is Peano. The former part is not very easy but the latter part is. For the proofs see Chapter 31 of General Topology by Stephen Willard.
It depends on your definition of arcwise-connectedness: in some books path-connected and arcwise-connected are the same. In other literature arcwise-connected is stronger since you require a continuous inverse. You can find more info here.