A centroid is the center point of given polygon feature. To find the centroids of your polygon layer and calculate the distance between these points, follow this procedure:
- Make sure your map is using a projected coordinate system.
- Open the attribute table of your polygon layer, and click on the button in the top left corner and select Add Fields...
- Add two fields named Cen_x and Cen_y, with the type as Double.
- In Cen_X, right click on the header and select Calculate Geometry.
- For Property, select "X Coordinate of Centroid", and select the units of your map.
- Repeat these two steps for Cen_y using "Y Coordinate of Centroid" as the Property.
- Click on Options > Export. Save the table as a DBF. Add the table to your map.
- Right click on the newly created table, choose Display XY Data, select Cen_Y for the X and Cen_Y for Y, and select the map's coordinate system. An events layer will be created; note that the events layer is not a shapefile, but instead was created on the fly.
- Open the Point Distance tool in ArcToolbox (Analysis Tools > Proximity > Point Distance). The event points are both the input and near features. Your output will be a DBF table of the distance between all points on the map.