To not have overlapping polygons/features, I normally use one or more polygons to clip/shape other polygons. This is often much faster than tracing when adding polygons etc. I cannot find this functionality in QGIS.
The highlighted polygon is overlapped by the adjacent (west/left) polygon with a straight edge. I want to cut the adjacent polygon with the highlighted.
You can substract (clip) a polygon from another of the same layer by using the plugin Clipper. You have to choose the polygon you want to clip and then you do Vector->Clipper->Clip and the plugin substracts the polygon from the bigger one. Then you choose the clipped polygon and delete it.
If you can't choose the polygon you really want because the QGIS chooses the other one, you can use the identify tool (i) and then choose the right polygon by using the Toggle Feature Selection from the Identify Window.
By doing this you can select the polygon you want when there is an overlay between features.
There is a plugin called Digitizing Tools that has the feature "Cut with polygon from another layer"...
In the image below I am editing the Green layer and have a polygon feature selected from the Purple layer.
When I click the 'cut with polygon from another layer' it will remove the portion from the Green layer that I had selected in the Purple layer:
I found the tool I was looking for! It can be done using the plugin called "Clipper".
You do not need to install any plugin to do that operation: use SAGA tools. Go to Processing>Tools (tool's pane opens on the right of your screen)>SAGA>Vector polygon tools>Polygon self-intersection. This tool will basically create a new layer composed of every possible combination of intersected polygons into the same layer.
