How do I change the Table of Contents in Word 2010 to show only headings 1-3?

If the table of contents has already been created, press Alt-F9 to display the mark up. In my case this was

{ TOC \o "1-6" \h \z \u }

Change 1-6 to 1-3, giving:

{ TOC \o "1-3" \h \z \u }

Then press Alt-F9 again to return to the normal view and update the table as usual.

1. On the ribbon, select References, Table of Contants, Insert Table of Contents...
2. Under General, set Show levels to 3.
3. If you want to select other styles to be shown on the TOC, press Options and type the level number next to the styles you want to include. This is useful if you have separate styles for chapter headings and apendicies.

In the Table of Contents dialog, just change the "Show levels" setting from 6 to 3. Reference here: http://office.microsoft.com/en-us/word-help/change-the-levels-in-a-table-of-contents-HA102322816.aspx