Fat is solid at room temperature; Oil is liquid.
Everyone needs some fat in their diet. Around 20-30% of daily calories.
A tablespoon of fat or oil has 120 calories! Or 250 calories per oz. or 9 calories per gram.

Vegetable Oil – from peanut, soya bean, sunflower, sesame, coconut, olive, and other vegetable oils
Animal Fat – lard (pig fat), fish oil, and butter. From fats in the milk, meat and under the skin of the animal
Hydrogenation – artificial process conversion of liquid vegetable oils to solid or semi-solid fats (as in margarine). turns unsaturated fat to saturated fat. Creates trans-fat. Increases the risk of heart disease. Very Bad.
Hydrogenated oil – vegetable oil that has been hydrogenated. Keep away.
Partially hydrogenated oil – vegetable oil that has been hydrogenated to some degree. Keep away as well.
Saturated fat – occurs naturally in animal fats, or artificially in vegetable oils when hydrogenated. Use in moderation – raises risk of heart disease
Monounsaturated fat – “good fat” – lowers bad blood cholesterol levels (LDL). May increase good cholesterol (HDL)
Polyunsaturated fat – “good fat” in moderation.
Trans-fat – “bad fat” created by artificial hydrogenation. Increases the risk of heart attack even in small quantities.
Cholesterol – found only in animal fats. Humans have cholesterol too, but it is mostly derived from saturated and trans-fats, not directly from animal cholesterol.
Omega-3 - polyunsaturated fat. Required in our diet. A quarter teaspoon a day. sources: leafy veggies, fish, fish oil, eggs, chicken.
Omega-6 – polyunsaturated fat. Required in our diet. sources: seed oils – soybean, safflower, sunflower or corn.
(Note: the right proportion between omega-6 and omega-3 intake is important. It should be 4:1, but in most western diets it is 10:1. That’s why we are all being encouraged to consume more omega-3.)
Omega-9 – polyunsaturated fat. good. sources: olive and nut oils.

Advanced:
Fatty acids – the building blocks of fat. The above terms refer to fatty acids (polyunsaturated fatty acid, omega-3 fatty acid, etc…)
Lipid – the scientific term for fat
Triglyceride – a combo of 3 fatty acids found in fat.

Sources:
What to Eat, Marion Nestle
American Heart Association