Partially hydrogenated oil is not the same as trans fat. The term "partially hydrogenated" means that hydrogen gas has been bubbled up into an oil to increase its degree of saturation and shelf life. The process of hydrogenation causes several chemical changes to occur in the oil. One of these changes (only one, but an important one) is the creation of trans fat.
There is some naturally occurring trans fat in many foods—but not nearly as much as we get from partially hydrogenated oils. Some of us get about 20 grams of trans fat per day solely from consumption of these oils. From natural foods, we would only get a few grams. A food label can claim "Zero grams of trans fat" even when there is partially hydrogenated oil in the product, because a product is legally considered "trans fat free" as long as there is less than 0.5 grams of trans fat per serving size. Trans fats are definitely harmful to our health in large amounts, and processed products containing hydrogenated oils are not ones that we recommend consuming.