Neutronium is the stuff neutron stars are made from. Unlike normal matter, it does not consist of atoms. It is composed solely of neutrons sitting right next to each other, like in an atom's nucleus.
The Similarities and Differences to an Atomic Nucleus
A normal atomic nucleus consists of one to a few hundred nucleons (protons and neutrons). Inside such a nucleus, there is a situation of balance between two fundamental forces of the universe: the electromagnetic force and the strong nuclear force.
The strong nuclear force is far more powerful than the electromagnetic, but only over short distances. It's what keeps the nucleus from falling apart, since all nucleons are attracted to each other by this force.
The electromagnetic force, on the other hand, makes the protons repel each other, because they all have a positive charge (just like the positive pole of a magnet). This force is weaker than the strong nuclear force, but it acts over longer distances. Now if a nucleus becomes very large (~200+ nucleons), the electromagnetic force tears it apart, resulting in nucleons being flung away (radioactive decay). Why? Because now, many nucleons are so far apart that they are no longer attracted to each other by the strong nuclear force. So the repulsion of the protons becomes dominant.
Neutronium, in contrast, does not contain any protons that would tear it apart if it became too big. That means that neutronium can reach sizes far beyond a mere 200 nucleons. It can contain just about any number of neutrons, while still remaining something like a single nucleus, without being torn apart.
Unlike atoms, neutronium does not contain any electrons. This is because in an atom, the electrons are attracted by the protons in the atom's nucleus (opposites attract: electrons have a negative charge), which causes them to orbit the nucleus. But since neutronium contains no protons, there are no electrons orbiting it.
What sets Neutronium apart from Normal Matter
It has a structure quite unlike normal matter, as it does not consist of atoms, or even molecules. Because it contains no protons, it is not radioactive.
Neutronium is totally chemically inert. This is because there are no electrons in it, which would be needed for chemical interaction.
Neutronium is probably very hard, since the forces holding it together are many times stronger than the forces between atoms and molecules. This is why neutronium is frequently found as a kind of armor in science fiction stories.
Most importantly, neutronium is incredibly dense, and therefore incredibly heavy. The density of neutronium is 4 * 10 ^ 17 kg/m^3 (four times ten to the power of seventeen kilogrammes per cubic meter). Compare that to the density of lead, which is a mere 1.134 * 10 ^ 4 kg/m^3. So neutronium is more than a million million times heavier than lead! That is why a neutron star of a few kilometres diameter has the same mass as our sun.
See The Sub-Atomic Physics Project and The Nuclear Physics Project for more information on the topics above.
To the physicists on E2: I admit to being somewhat imprecise in this writeup. I know that it's actually far more complicated than that, and I've read that there seem to be protons in a neutron star as well. However, I think that this level of complexity completely suffices.
Here's how I calculated the density of neutronium:
neutron radius = r = 10^-15 m
neutron mass = m = 1.6749*10^-27 kg
neutron volume = V = 4/3*pi*r^3 = 4.18879*10^-45 m^3
neutron/neutronium density = m/V = 4*10^17 kg/m^3
/msg me if I made an error somewhere.
Sources: mostly from my head, some from "Materie in Raum und Zeit" by Sexl, Raab and Streeruwitz, Sauerländer Verlag, ISBN-3-7941-4041-9 | 
