You need to use two features of Word to be able to achieve this namely "Different odd and even headers" and "Sections".
For having different headers on even and odd page, you can select the option different even and odd page headers. In Word 2003 and earlier, after you go to View->Header and Footer, click on "Page Setup" and select "Layout Tab". Under "Headers and Footers" select "Different Odd and Even". Then you will be able to enter one header for odd pages and another for even.
Since you want to use chapter name in the header, you will need different headers for different parts of the book. To achieve this, you need to divide the document into sections. To know how to create and use sections, please refer to this article.