The two main properties that make steel interesting are 1) cost 2) toughness.

Toughness in a material science sense is the ability to absorb energy before failure. This has two major components, yield strength (energy to initiate deformation) and ductility (ability to deform without fracture).

The ductility of steel (and of many other metals) is what drive their use in structural applications, whereas many materials, i.e. ceramics, have higher yield strength but almost no ductility. Even though many glasses are "stronger than steel", if you drop a glass bowl and a steel bowl only one will shatter. If your I-beam shatters you are in trouble.

Steel is interesting in comparison to other metals because iron and carbon are abundant and the iron-carbon system has a lot of interesting features that can increase both strength and ductility.

Well, its elastic modulus and yield strength are pretty important, actually. Being stiff and strong is pretty fundamental to a lot of its uses. It's stiffer and stronger than most other everyday materials and virtually all everyday plastic materials. Glass, quartz, alumina, zirconia, and porcelain are stiffer than steel, and glass can be stronger, but they're all brittle rather than plastic, which is pretty inconvenient and often makes them very weak in tension. Wood, most other fired clay, aluminum, brass, nearly all organic polymers, mica, cotton, dirt, etc., are much floppier and weaker than steel.

But there are a lot of metals that are somewhat stiffer and stronger than steel, like chromium, platinum, and tungsten, while still being somewhat plastic. The great advantage that steel has over them is that it's unbelievably cheap. It's even cheaper than brass, bronze, and lead!

Plasticity (ductility and malleability) is important for a couple of reasons. First, as I mentioned above, it greatly increases the fraction of the material's theoretical strength you can get in practice. Second, it allows you to form the material instead of cutting it to shape. That's the property you're using when you wrap a sandwich in aluminum foil or tie a gate shut with baling wire. You can't do that with porcelain foil or porcelain rod. Third, ductile failure happens gradually rather than suddenly, which is important in some cases.

The other really interesting thing about steel is that it's hardenable. This is very significant because cutting and forming hard things is hard. So it's routine to cut or form steel in its soft state to get more or less the shape you want, harden it, and then grind it and maybe lap it to the precise shape you want. Grinding and especially lapping can be very precise and cut very hard materials, but they're very slow processes.

Finally, steel can withstand much higher temperatures than organic materials, or even most other common metals.

These are, I think, the major reason why steel has so extensively displaced what Andrew Carnegie liked to call "inferior materials".

Mainly it's the ability to absorb energy and punishment. Many materials have an elastic range, where if you deform it within that range, it will return to its original shape. Of those, a lot of materials, including many metals, are relatively brittle - once you push it past its "elastic limit" it just breaks. There are ways to make steel more like that too, but in general steel doesn't break at that point, it "yields" or deforms permanently (inelastically). You can keep on deforming it (and in fact it actually gets slightly stronger while you're doing that, which is an interesting feature) and it goes way way beyond what you would think possible before it finally breaks. So as a result it can absorb tons of energy, which makes it interesting for strength applications. The steel frame of a building in an earthquake for 30 seconds is absorbing tons of energy while hopefully not collapsing, or even in the worst case it at least allows a bunch of extra time for at least some of the occupants to escape. Or a steel-framed car that crashes into a pole - the steel crumples, absorbs energy, and slows the car somewhat more gradually in the process. If the frame were a brittle material it might just shatter on impact.

It's dirt cheap and incredible versatile. It has many different crystal structures that can give you an incredibly wide range of properties. General purpose mild steel used in buildings. Stainless steel that resists corrosion. Maraging steels used in aerospace.

Also, the high modulus is interesting. Some components are stiffness limited such that you couldn't use aluminum or titanium even if you wanted.

It's really tough (i.e. not just strong but ductile as well i.e. able to stretch more without fracturing like a brittle material would) and relatively cheap to manufacture (both because the industrial process is scalable, and because iron is incredibly abundant). When cost isn't an issue, you don't use normal low alloy steel - you use expensive alloying elements like chromium for corrosion-resistant stainless steel or more relevant to this comparison, nickel for maraging steel which is several times as strong as ordinary steel.

Some great answers here, but one other thing to add, is that it's also a material that's quite easy to recycle.

another great plus is its weldable