Steel has some carbon content. How much carbon basically determines the grade of steel. Mild steel has a relatively low carbon content, spring steels and tool steels have a higher carbon content. The amount of carbon also determines how hardenable the steel is. Mild steel cannot be hardened at all other than for case hardening. Spring steels and tool steels can be hardened... some of them excessively so, but they tend to be less tough and more brittle when they are hardened. This is why it must be tempered for most applications if it has been hardened.
Pure iron is not normally encountered outside of a laboratory. It is iron only and has no carbon. Wrought iron has been called pure iron, but it is not really pure. See the next paragraph.
Wrought iron has pretty much been out of production since the 1940's when the last of the wrought iron pipe ceased to be made. It has no carbon content other than perhaps trace amounts. It contains bands of slag, so it is easily recognized as it seems to have "grain" just like a piece of wood. It does have a few useful attributes in that it is highly resistant to rust and it can be welded in a coal or charcoal forge quite easily. You don't even absolutely need to use flux for welding it in the forge. It is much softer than steel though. This is one reason that the original Colt revolvers are black powder only. Their frames and maybe cylinders (I forget) were made of forged wrought iron.
Cast iron has a very high carbon content, so high that it is too brittle for forging, hence the name cast iron. There is also cast steel, which is another animal entirely.
Bear in mind also, that before the early 1900's, nobody really knew much about making steel. It's characteristics varied batch to batch, so if you got an antique gun allegedly made of steel, just remember that all steel is not created equal.
Now, to further confuse you, there are other things than carbon that steel is often alloyed with for different applications. These alloys can all be basically grouped as mild steel, medium carbon steel/spring steel and tool steel. Each variation has different attributes and if you are buying it from the dealer, you can get all the specs you need on it from him. Most hobby smiths work with primarily mild steel which can be either bought or salvaged. When we find actual wrought iron, we squirrel it away like a miser hoards gold.
This is the simplistic version, basically Steel 101 in a single paragraph. If you want more detail, you need to either visit a library or maybe go drinking with a metalurgist, a blacksmith and a modern welder. These types of folks can tell you more about steel than you actually want or need to know. They can tell you about Wootz steel, crucible steel, Damascus steel and probably some steels I never heard of.
As with everything, your mileage may vary.
