There are two answers to your question depending upon whether you are using Excel 2007/2010 or Excel 2013.
Unfortunately you cannot insert new columns into an Excel 2007/2010 pivot table.
You can, however, copy the pivot table and paste it "as values", which will extract the data from the pivot table and allow you to edit it in any way you wish (but, of course, it will then no longer be a pivot table)
For full instructions on how to paste "as values", see Essential Skills Lesson 3-5: Use Paste Values and increase decrease decimal places displayed.
For instructions on how to create a pivot table, see Expert Skills Lesson 5-1: Create a pivot table.
This version of Excel introduces a brand new Data Model feature covered in Expert Skills Session Six: The Data Model, OLAP, MDX and BI.
Pivot tables based upon a Data Model look very similar to those based on a single range but actually work in a different way and have different features. Their primary feature is the ability to reference more than one source data table a the same time.
Without going too far into the details, a Data Model based pivot table allows you to convert the pivot table into a set of formulas that directly reference the data behind the pivot table. In this case the (converted) pivot table is still a pivot table but you can insert columns, cut, copy and paste values (that still reference the pivot data) and also format the pivot table in any imaginable way.
There are several new concepts associated with the Data Model that you need to learn so we have devoted an entire session to the Data Model in the Excel 2013 Expert Skills course.
If you need any more assistance with this, please feel free to reply to this post.