INCI stands for International Nomenclature of Cosmetic Ingredients. It's the FDA approved way to list cosmetic ingredients, so this is probably how you'll see this ingredient listed on a product.
Saponified literally means "turned into soap". This is the name of the process that turns fats and oils into soap by using an alkali. Old-fashioned soap makers used to use wood ash as an alkali, but we have much more pure sources of it today.
Essential oil are distinct from fragrance oil. Essential oil are extracted directly from the plant (the leaves, fruit, or other parts of the plant), while fragrance oil are synthetic compounds designed to mimic a particular scent. We've found that people with skin sensitivities are sometimes sensitive to fragrance oils (because there can be several hundred components in them), but are much less likely to be sensitive to essential oil.
Extracts are used when obtaining an essential oil is not possible or practical. This can be the case when the oil is not easily obtained from a plant, or a plant doesn't have a sufficient amount of natural oil to harvest. To create an extract, plant parts (leaves, flowers, roots, or stems) are macerated and soaked in a solvent such as water or alcohol.