How to fix overlapping polygons using ArcGIS for Desktop?

the quick and dirty method consists in using the "union" tool with the layer alone. Two new polygons will be created where polygons are overlapping and you can remove one of them. This is not straightforward but you can use "find identical" to get the polygons to be deleted.

The more advanced solution consists in building a topology for your layer (with the mustn't overlap" rule), then you can using the topology fixing tools.

Subtract: The Subtract fix removes the overlapping portion of geometry from each feature that is causing the error and leaves a gap or void in its place. This fix can be applied to one or more selected Must Not Overlap errors.
Merge: The Merge fix adds the portion of overlap from one feature and subtracts it from the others that are violating the rule. You need to pick the feature that receives the portion of overlap using the Merge dialog box. This fix can be applied to one Must Not Overlap error only.
Create Feature: The Create Feature fix creates a new polygon feature out of the error shape and removes the portion of overlap from each of the features, causing the error to create a planar representation of the feature geometry. This fix can be applied to one or more selected Must Not Overlap errors.