Butter, margarine, oil
Butter is made by churning cream or milk, to be called butter should consist of mostly milk fats. More generally it could refer to any edible fat that is solid at room temperature.
Clarified butter consists of just the fat, which was separated from the milk solids and water through rendering.
Clarified butter can be used at somewhat higher temperatures (has a higher smoke point), making it interesting for sautéing and such.
Margarine is an emulsion of water and oils, be they animal or plant-based. It historically was a butter alternative as it is cheaper, and can be kept longer.
The fact that margarine contains a good deal of water makes it less fit for baking as it is more likely to spray hot fat as the water evaporates.
People may regularly refer to margarine as butter, although in many countries the packaging cannot legally do so.
Vegetable margarine and soy margarine is vegetarian and vegan-safe (usually/always?).
Vegetable oils are often based on things like canola, safflower, sunflower, corn, soybean, peanut, sesame and such.
Products like margarine and vegetable oils are also interesting to vegetarians, vegans, and people with allergies, because they usually use a specific ingredient and avoid others.
Note that specific margarines use no cream, buttermilk, or whey, or so little that the lactose lactose intolerant will not be bothered by it.
You may want to specifically look (or avoid) for nut oils, (vegetable) margarine, soy oils, etc.
There are also kosher margarines.
Shortening
Shortening refers to fats used in cooking that are usually solid at room temperature, and have a higher smoke point than butter or margarine.
In a broader sense it can refer to margarine, lard, etc., but is often specificially refers to pure fat (often with little or no taste), and is often specifically used to give bread and such a more crumbly texture (it avoids long protein strands in wheat-based doughs, although that is not the origin of the term).
Shortening may be butter, lard, margarine, but can also be purely vegetarian (and particularly earlier shortening was not so healthy).