When you want to make a solenoid core, you are typically interested in a material which exhibits low eddy current losses (eddy currents are induced during a change in field, and they result in heating / losses in the core - this is almost never desirable except in the case of induction heating), and with high permeability - the latter enhances the magnetic field. Finally, you want high saturation: to be able to make a strong magnetic field, you want the material to remain linear (the more the atoms align in the field, the stronger the over all induced field becomes).
The second and third of these reasons make soft iron a good material for DC electromagnets: you can get a very high field from them. However, the conductivity of soft iron results in eddy current heating when it is used in an AC application - for example transformers. For this reason, it is usually laminated (cut into thin slabs or lamina which are electrically insulated from each other). This makes it harder to set up eddy currents, and limits the heating.
For permanent magnets, you are mostly interested in the remanence: how much magnetic field remains when you remove the driving magnetic force (usually an electromagnet). The coercivity tells you how much (reverse) magnetic field would have to be applied to demagnetize the material. So the former says "how strong can be magnet become", and the latter says "what will stop it being a magnet". A third consideration with some permanent magnet applications is their mass - so you might be interested in these properties as a function of mass.
For permanent magnets, Samarium Cobalt ( $ {Sm_2Co_{17}}$ ) and Neodymium Iron Boride ($ Nd_2Fe_{14}B $) are among the strongest. They are called "magnetically hard" because of their high remanence and coercivity. You would not want to use them in transformers etc. because there you need the induced flux to follow the current. The hysteresis in their magnetization curve would give rise to excessive heating during each magnetization cycle.
The difference is probably best explained with this picture: from ttp://www.daviddarling.info/images/types_of_hysteresis_loop.jpg
The larger area under the red curve shows that the material holds its magnetization well - but also that a lot of work has to be done to change the direction of the field. By contrast the blue curve shows that soft iron doesn't need a large applied field to lose its magnetization - its coercivity (the horizontal width of the curve at Y=0) is low.
The are various crystal forms that iron and steel can adopt, the common ones being ferritic, martensitic and austenitic. The ferritic and martensitic forms are ferromagnetic (or just magnetic in everyday terms) while the austenitic form is not. So it isn't simply that iron is magnetic and steel isn't, it is specifically austenitic iron and steel that isn't magnetic. However things may not be as simple as this since lumps of steel may well contain grains of more than one crystal type, so they may be partially magnetic.
Now you're going to ask me why the austenitic crystal form isn't magnetic, and I don't think there is a good answer for that. Ferromagnetism is actually quite a subtle effect dependant on exactly how the electrons in the iron atoms interact. It's due to a quantum mechanical phenomenon called the exchange interaction. In the ferritic and martensitic crystals this effect is large enough to make the spins line up and generate macroscopically ordered domains. In the austenitic crystal it isn't. I don't know of an intuitive way to explain why.
Best Answer
The short answer is, yes, it can be modified.
The coercitive field depends a lot on the microscopic details, such as grain boundaries and the material composition. For strong changes in the magnetic properties you need additional elements, such as Nb, Co or Ni. Additionally the details of the manufacturing process can influence the magnetic properties as well (sintering parameters, annealing).
A nice introduction into different materials and optimization possibilities can be downloaded from Vacuumschmelze, a large manufacturer of permanent magnets.