Low Carbon Steel or Mild Steel. It have between 0.05 and 0.25% of carbon. It have lower strength than other form of steel, but it's also more ductile which mean it's easier to machine or work with. This make that steel ideal for structure like building, bridge, etc. Because even if they have a lower strength, they are easy to shape into different shapes. The fact that it's ductile make is also safer in building, basically the building will bend which give notice that something is weird and it won't just collapse on the people right away.

High Carbon Steel have between 0.61 and 1.5% carbon. This type of steel is very strong, very hard and harder to machine or form into shape. This make it ideal for item that will have to endure a lot of wear like tools.

Medium Carbon Steel have between 0.26 and 0.6% carbon. It's an happy medium between lower and high carbon steel. Things like gears, bolts, automotive components, etc are usually made of that steel. They are parts that will suffer some regular wear so mild steel would be too damage, but it doesn't suffer enough wear to justify the higher cost and difficulty of working with high carbon steel.

Cast Iron have between 1.7 and 3% carbon. It's much harder than Steel, but also very brittle. Mild Steel usually around 130HB of hardness, medium carbon steel is around 200HB, but cast iron is closer to 400HB.

Stainless Steel have a high content of chromium (between 10 and 30% depending the type), but it can have different level of carbon up to 1.2%. That said, just like normal steel, the most used type of stainless steel will have lower amount of carbon than that.

Alloy steel well those can be anything, different type of metal added to the steel (usually in small percentage) will have different effect. Molybdenum increase the toughness so some cutting tools, turbine blade, rocket motors' parts, etc. Tungsten increase the melting point. Boron increase the hardness. Bismuth make it easier to machine. And many more.

Wrought Iron have a very low carbon, less than 0.08%. As you can see from 0.05-0.08% it could be both wrought iron or low carbon steel. That's because wrought iron isn't define by the carbon content, but by how it was produced. See in the past we were not able to measure the amount of carbon, so we didn't categorized iron by their content like we do today, but rather by the way we produced it. Wrought Iron (low carbon) was made from iron ore in a bloomery, but it take a long time. Pig Iron (high carbon) was produce by using a blast furnace and it was faster to produce than Wrought Iron. What you could do after is take your molten pig iron and add stuff like coke, limestone, you can also burn the impurity and you end up with cast iron. Or you can take your pig iron into a finery forge or puddling furnace to make wrought iron, this was faster than the earlier bloomery. Other things we used to do was beat the hell out of cast iron to slowly remove the carbon out of it or mix cast iron with wrought iron to reach a better level of carbon. Both medium were relatively expensive at the time.

It's only later when we develop steel metallurgy that steel took over the traditional iron, and now we categorize steel by their content, rather than the method of production. But now since both way of categorizing exist, it can be a bit confusing.