The Erase Tool allows you to erase parts of paths which you no longer need. You cannot use this tool with text paths, i.e., paths on you which you typed using the Type on a Path Tool.
Here is how you use it:
- Select a path with the Direct Selection Tool
- Select the Erase Tool.
- Drag your Erase tool, with your mouse button pressed, along the part of the path you want to erase (not across the path).
- Release the mouse button and you are done.