When you click on an anchor point with the Direct selection tool, you select the two path segments on either side of the anchor point (unless of course it is an endpoint, in which case you will select the single path segment connected to that anchor point).
In the following demonstration movie, we will change the stroke colour of a circle so that the top half has a red stroke and the bottom half a blue stroke. To do this, we perform the following steps.
Click on the Direct Selection tool to make it the active tool.
Click on the top anchor point of the circle to select the two segments on either side of it, thus selecting the top half of the circle.
Choose Edit - Cut.
Choose Edit - Paste in Front (or Paste in Back. Since the original object has been cut, either command will do.)
Choose blue as the stroke colour.
Click on the bottom anchor point of the original circle (what's left of it).
