How to create a footer with a chapter name and page number?

The best way to automatically do this is through Fields. Assuming the format of your chapters or headings are styles, you need only to add a reference to styles.

In Word 2010, while editing the Header & Footer go to Header & Footer Tools → Design → Quick Parts → Field....

In Word 2013 it's in Insert → *Quick Parts → Field → StyleRef (thanks to @techbrunch)

Now in the Field dialog box, choose StyleRef from the Field name list and your desired heading from the Style name list.

You can similarly add page numbers by choosing Page in the Field name list.

Additional Notes

Using fields, the data is automatically updated, and you don't need to repeat anything per different pages or sections. In pre-ribbon Word versions (2003 and earlier) simply go to Insert → Field when editing the header/footer.