There are a couple of ways that you have to control the display the index of a web part in browse mode.
First, the index is only shown if you have 2 or more web parts with the same Title. The easiest way, and probably a best-practice, is to give each web part a distinct title. Even if you are using the same web part more than once on a page. Chance are each instance shows different information and should have a unique title.
Second, you can set the Web Part's Chrome type property to display: No Chrome :None, or just a border around the web part: Border Only. These settings will not only remove the web part index but the whole title. I used these settings when I'm using Web Parts to assemble a page and I want it to look like a more traditional web page.
That's about it, these are the only two choices you have to remove the index added to the title of Web Part's with the same Title.