Great question. Emboss is a rarely used word. It's used mainly in design. It means you create a pattern that is raised above the flat portion of the surface. Best example I can think of right now is let's say you have a bug bite and your skin swells. It raises the skin above the rest so this is similar if you emboss something. Engrave is the opposite. You make a depression - an area that is below the rest of the surface. It is also used usually if you put your name or a logo or something meaningful to you into an item. Like let's say you have a wedding ring, you might want to engrave your marriage date or some phrase into it. Carve is usually used in a cruder sense. And it might also involve removing an entire area of a surface to make a shape. You might carve some pattern into a tree. In United States, one popular example is "carving a pumpkin" for Halloween. Here we take a pumpkin and carve out portions of it to make a scary face.

In one way they all mean the same thing, the removal or alteration of a material to make a mark. The word choice will depend on the technique being used to make the marks. In carving you would normally use a knife or a sharp edge to gouge or cut out pieces, you would carve wood for example. In engraving you would use a thin rotating metal bit to cut out the material, it can produce very intricate marks. You would engrave your name in metal for example Embossing uses a press system, where two plates, normally metal, would put a material under pressure creating an image. You would emboss paper, leather or even thin metal for example. It provides a simple way to reproduce the same image.