Why are magnets magnetic - what makes things magnetic?
The basic ingredients of any matter are atoms. Each atom is made of a nucleus which contains protons and neutrons, and electrons that orbit the nucleus. The electrons have two important features: first, they all carry charge. Second, they are constantly in motion. In fact, their motion can be divided into two parts. The first is their orbit around the nucleus, and the second is intrinsic rotation, known as "spin". This is analogue to the earth's motion: the earth both orbits the sun (completes one orbit in a year) and, at the same time, rotates around its own axis (completing one rotation in 24 hours =1 day).
Now, here comes the trick: when an electric charge moves in a closed loop, it creates a magnetic dipole. This means that every electron can be viewed as a tiny magnet, with both north and south poles ("above" and "below" its orbit). Due to the electron's intrinsic motion (spin), in fact the electron doesn't even have to be attached to an atom in order to produce the magnetic dipole.
Now, each atom contains many electrons. However, according to the laws of quantum mechanics, these electrons can only orbit the atoms in certain orbits (in other words, the electrons orbits are quantized). These orbits are known as "shells". In each shell, there can be a maximum number of electrons: e.g., in the innermost shell there can be maximum 2 electrons, in the second shell 8, etc. If an atom has more electrons than can be in a shell, the additional electrons have to reside in a different shell.
If a shell is completely filled with electrons, the electrons will necessarily have opposite spins, and as they carry the same charge, their magnetic dipoles will be in opposite directions - and cancel each other. However, in cases where the spins are aligned in the same direction, which can happen in those atoms in which the shells are not fully filled, the magnetic dipoles add up, resulting in a net magnetic moment of the atom.
Materials in which the dipoles add up spontaneously are known as "ferromagnets". This happens provided these materials are cold enough (below a temperature known as "Curie temperature"), otherwise the electrons orbits become random - and the net magnetization disappears. A classical ferromagnetic material is of course iron, for which the Curie temperature is around 1000 degrees [Kelvin]; thus, in room temperature (300 degrees Kelvin, far below this value), iron will have a magnetic dipole, which most people will simply call it as "magnetic".