Manufacturing is the process of turning various items into a combined product. The resulting product may or may not be more than one, depending on the manufactured item itself (for instance Sesame Oil is produced in groups of 2).
To manufacture an item, start by opening up the "Make" window (open your inventory all the way, the button is on the left side near the middle). After that you need to place each of the required items into the make window (left click once on the item, then left click again where you want it to appear). Once the items are in place, click on the left drop down menu to choose what item to make from those materials. After choosing, type the number of items you want to manufacture in the second box. Finally, click the small arrow to begin manufacturing. Remember that only the required materials will be used when making an item, not every item that has been placed in the "Make" window.
When manufacturing items, it's good to be aware that some require manufacturing skills and that the majority of these have a failure rate that varies depending on your current manufacturing skill level for that type of item and the specific item. Japanese Sandals, for example, can be made with no basic skill in Shoe Manufacturing, however Straw Sandals requires a minimum of 5 skill to produce it. Additionally, a skill of 5 will have a low success rate, whereas having a skill of 10 would bring the rate up to 100% (even if it's above 100%, there are no additional bonuses).