A ferromagnetic material is one that has magnetic properties similar to those of iron. In other words, you can make a magnet out of it. Some other ferromagnetic materials are nickel, cobalt, and alnico, an aluminum-nickel-cobalt alloy.
Magnetic fields come from currents. This is true even in ferromagnetic materials; their magnetic properties come from the motion of electrons in the atoms. Each electron has a spin. This is a quantum mechanical phenomenon that is difficult to make a comparison to, but can be thought of as similar to the rotation of the Earth about its axis.
Electron spins are in one of two states, up or down. This is another way of stating that the magnetic quantum number can be +1/2 or -1/2. Electrons are arranged in shells and orbitals in an atom. If they fill the orbitals so that there are more spins pointing up than down (or vice versa), each atom will act like a tiny magnet.
That's not the whole picture, however; in non-magnetic materials such as aluminum, neighboring atoms do not align themselves with each other or with an external magnetic field. In ferromagnetic materials, the spins of neighboring atoms do align (through a quantum effect known as exchange coupling), resulting in small (a tenth of a millimeter, or less) neighborhoods called domains where all the spins are aligned. When a piece of unmagnetized iron (or other ferromagnetic material) is exposed to an external magnetic field, two things happen. First, the direction of magnetization (the way the spins point) of each domain will tend to shift towards the direction of the field. Secondly, domains which are aligned with the field will expand to take over regions occupied by domains aligned opposite to the field. This is what is meant by magnetizing a piece of iron.
Iron comes in two forms, hard and soft. If you were hit on the head with a soft iron bar, it would still feel very hard; soft is simply a term describing the magnetic properties. In hard iron, the domains will not shift back to their starting points when the field is taken away. In soft iron, the domains return to being randomly aligned when the field is removed.
Hard iron is used in permanent magnets. To make a permanent magnet, a piece of hard iron is placed in a magnetic field. The domains align with the field, and retain a good deal of that alignment when the field is removed, resulting in a magnet.
An electromagnet, in contrast, uses soft iron; this allows the field to be turned on and off. It's easy to make an electromagnet. One method is to coil a wire around a nail (made of iron or steel), and connect the two ends of the wire to a battery. A coil of wire with a current running through it acts as a magnet all by itself, so why is the nail necessary? The answer is that when the domains in the nail align with the field produced by the current, the magnetic field is magnified by a large factor, typically by 100 - 1000 times.
Magnetic effects are sensitive to temperature. It is much easier to keep permanent magnets magnetized at low temperatures, because at higher temperatures the atoms tend to move around much more, throwing the spins out of alignment. Above a critical temperature known as the Curie temperature, ferromagnets lose their ferromagnetic properties.