I need for my table to run horizontally, not vertically. In other words, I would like to have the header on the left-hand side, rather than on the top. I am working in InDesign CS6.
If you are trying to achieve what @Lucian shows above, you won't be able to do that and use the Header Row feature. Just create a table with the appropriate number of columns and format the leftmost column to look like a header. If your table spans multiple pages, you will have to manually create and format a left hand "header column" for each new page.
If you are looking to create a table where the text is actually rotated, so that you need to turn the page 90 degrees to read the text, then all you have to do is select the cells with your text tool, and on the ribbon at the top, you should see the option to change text direction.
If this is what you are trying to achieve, I suggest going to the View menu, and choosing "Rotate Spread." This will rotate the page that you are working on so that you are working normally, but does not change the orientation of the actual document page. Saves wear and tear on your neck!