Q:

How to extract polygons that are not in one layer and overlap maybe a little bit

I have two layers.   Some polygons overlap and some have new polygons that are not in my main layer however they overlap at some intersections.  How can i select and extract only the polygons that are not in my main layer and have this data as a shape file or feature class. IE get only the polygons that are on my second layer which has additions to the main layer basically. 

A:

For clarity, let's call your two polygon layers Green and Blue.   If I understand correctly, you want to:

Select those Blue polygons that DO NOT intersect any Green polygons. 
Export the selected Blue polygons to a third layer

To select the Blue polygons, I would first use the Selection > Select By Location tool, with Blue as the Target layer, Green as the Source layer, and intersect the source layer feature as the Spatial selection method.   The intersecting Blue polygons are now highlighted.   Next, right-click the Blue layer in the Table of Contents and drop down to Selection > Switch Selection.   This has the effect of selecting those Blue polygons that DO NOT intersect Green polygons, which is what you want. 
In the Table of Contents, right-click the Blue layer and drop down to Data > Export Data.  Enter the output path and name.   Voila! -  the third layer, containing only the intersecting Blue polygons, is created. 