Let's start with a die (as in dice). It's a cube with four corners and 6 faces with dots. Even if we made a blank die, we still have the 4 corners and 6 faces.

Iron, or ferrite, has a crystalline structure just like this blank die, with one atom at each corner forming a cube. Since it's also hollow inside that cube, there's one more packed in atom. This is called a body centered cube.

Now I take that die and I heat it up until it's white hot in a fire and the energy moves things around and the little atom in the middle pops out when the cube expands and vibrates. Now it settles back on the face of the cube making a die with just one dot on one face. This is austenite, and the structure is face centric now.

We can pack one atom on each face making dice with all one faces, which is awesome! With all of those atoms, the inside of the cube is too small for another iron so it's empty for the moment. When I toss that dice back into the fire though, a smaller carbon atom is able to slip in past the one dot faces and fill the center. Now we have an alloy of iron and carbon.

If threw a dense box of iron dice all neatly packed into the fire, the whole thing can heat up and change from ferrite (body centric) to austenite (face centric) but the carbon can only penetrate so far into the box because it has to move from one die center to another. This is basically "carburizing" a steel billet. If we fill every single die with carbon, the box will be too full and break open, so there's a sweet spot of how much carbon we want.

The way we made steel for a long time was to blow hot air into "the box" of molten dice and introduce lots of carbon into the iron, making "pig iron." Then the pigs were remelted without carbon present, ejecting lots of carbon as the cubes opened up aiming for some percentage of carbon between 1 and 3%.

There's is absolutely a difference between all of the percentages. I think of them as low carbon (mild) steel, high carbon, and cast; but this is a huge oversimplification. Cast alone has ductile, white, and grey varieties each with different properties. High carbon could be 0, A, W, D, white/blue/hitachi varieties again with different properties. Low carbon steel could be wrought or mild. Wrought iron has lots of silica in it as a flux agent and is no longer really made, but is desirable for it's properties to the right person.

As for ductility, no pressure on knowing that term specifically. I know you understand the concepts. In my mind, you could think of ductility as the ability to stretch without tearing like pizza dough, you want a really glutenous dough to make a big pizza. Malleability is a quality referring to bendyness, spaghetti is not malleable until it's cooked and then it's very malleable. Also, machinability is the ability to cut it down cleanly, room temp kerrygold butter has supreme machinability, ice has no machinability. Weldability is another quality, a gummy bear can have it's head reattached with a lick, but you can't put a sliced apple together again.

As an aspiring metalworker who forges, smelts, machines, and welds... I think about this stuff a lot.