When you're editing a road and hover over another road, you'll see the other road light up in blue. The cursor changes to get a little --+ logo, and at that point if you click, you'll link the edited way to this other one.
When editing roads that don't quite join, I use this sequence:
1: Select the end node of my road.
2: Drag it clear of the junction.
3: Click it again and you'll get the elastic 'more of this way' line.
4: Click in the target road where you want the junction. Click a second time to set the node and release the cursor.
(you'll now have the roads joining correctly, with a gratuitous extra node knuckling off somewhere nearby)
5: Select the gratuitous node and press [Del] to delete it, or move it to make a road curve, as appropriate.
Incidentally when editing in Potlatch, this is useful: To start a new way from the middle of an existing one, highlight the existing way, then hold down [Shift] while clicking the node. You'll get a ready-joined new elastic way.