Why does current change iron into magnet? When we pass electricity to an iron, why does it develop poles and change into a magnet? 
That is to say, on passing current around a coiled wire, it forms a magnet. 
 A: This follows directly from Ampère's law: 
$$\vec \nabla \times \vec H - \partial \vec D/\partial t = \vec J$$
Which under the assumption that the system is static becomes:
$$\vec \nabla \times \vec H = \vec J$$
To get rid of the "annoying" vectors we should integrate over a circular surface that is orthogonal to the wire (cfr the image below, where the surface is bounded by one of the blue circles):
$$\int d A \cdot\vec n (\vec \nabla \times \vec H) = \int dA \vec n \cdot\vec J$$
Where $\vec n$ is the unit vector along the wire.

The left hand side is easy, it is the flux density that is passing trough the entire surface. This is off course equal to the current that is flowing trough the wire: $\mathrm{rhs} = I$
The right hand side can be rewritten by application of the Kelvin-Stokes theorem and we find that:
$$\mathrm{lhs} = \int d\vec l \cdot\vec H  = 2\pi r H$$
Equating lhs to rhs yields: $$H = \frac{I}{2\pi r}$$  which is the well known formula for the magnetic field due current in a wire.
A: Why - questions are always tricky in physics. You first need to agree on a set of basic concepts/laws which you accept as true. There is no (mathematical) proof for any of these laws and usually they are postulated, and confirmed to be true by experiment. Whatever concepts/laws you accept as true, a very inquisitive person could always as "Why are these concepts/laws as they are?" This can lead you to an answer in terms of other, more fundamental concepts/laws, to which you could get another "why" question, etc.
Richard Feynman explained this much better in this video
For your specific example you could think of several explanations, in increasing level of generality (or you could say in chronological order of the history of physics; or you could think of how a child would learn about this effect as he/she grows up). This is not complete and you could add any number of intermediate descriptions/explanations:


*

*Attaching a battery to a piece of iron, makes a compass needle move.(Note that this is a very specific experiment and does not use concepts like "electricity, magnetic field,...". As an explanation this is only useful for this one type of experiment.)

*If, e.g. through other experiments, you realize (and accept) that there is such a thing as electricity, electric current, magnetic fields, and also realize how a compass needle acts to an external magnetic field qualitatively, you can get a slightly more sophisticated explanation, saying that the battery creates an electric current, the current creates a magnetic field and the magnetic field makes the compass needle move. While it does not answer your question, why the current creates a magnetic field, it is more general than the first explanation. For instance at this level you will be able to explain experiments where you have electric currents that are not in iron, or you can explain experiments where you don't have a compass (but for instance two wires with current).

*If you accept Maxwell's equations (or at least Ampère's law), you have a quantitative explanation as presented in gertian's answer. For all practical purposes this is all you need to know in order to explain this specific and many other experiments quantitatively and qualitatively.

*Physicists (at least some of them) tend to look for fundamental, more general laws from which other laws can be derived. Special relativity connects electric and magnetic fields. So at this level you can get rid of magnetic fields (or of electric fields) altogether  and describe everything in terms of electric fields (or of magnetic fields) only and transformations between reference frames. Still you need to accept the existence of electric fields (or of magnetic fields).

A: A piece of iron consists of many tiny randomly oriented crystals.  Within each the atomic magnetic dipoles tend to align themselves with the crystal axes.  When an external magnetic field is applied, as from a current carrying coil, the dipoles align with the external field.  When the external field is removed, the dipoles go back into alignment with the crystal axes but many of the dipoles may have had their alignment reversed. The result is a piece of iron with more dipoles pointing in one direction than the other producing a net magnetic field.
