Unable to bevel the selected edge. How to fix?

Bevel works best with clean manifold geometry where each edge has exactly two (or less) faces and that there are no reversed or internal faces.

If your model does not meet this criteria, then errors could occur when attempting to bevel an edge and you might get an error as shown below:

It may not always be obvious what is causing the error. But try doing the following steps:

Run the Clean Edges command and ensure all edges in the active context have been cleaned up.
Check your model carefully for reversed faces, especially near the edges you are trying to bevel. Flip any reversed faces (Right click, Reverse Faces) or try running the built-in SketchUp Orient Faces command (Right click a face and choose 'Orient Faces')
Check your model carefully for hidden internal faces and delete them, especially near the edges you are trying to bevel. The Section tool can be used to find internal faces as shown below (be sure to disable Section Fill in your Style settings)
If the error still remains after following these steps, it might be best to delete and re-model the portion that you are trying to Bevel. Be sure to keep the model clean and the faces oriented properly.