The world's strongest permanent magnets are called rare earth magnets. They are made of "rare earth" elements, such as Neodynium and Cobalt, which can be found in the bottom center of the periodic table. These elements have large magnetic moments. The magnetic moment of an atom comes from the spins of the electrons orbiting the atom.

Atoms with very high magnetic moments have unpaired electrons whose spins can align to give the entire atom magnetism, or a magnetic moment. When many atoms like this align their spins, we get a strong magnet.

In the case of rare earth magnets, elements with strong magnetic moments are combined (sometimes with metals like iron) and melted. The molten metal is molded, and as the metal cools it is subjected to a very large magnetic field. This external magnetic field causes the magnetic moments of the atoms to align. Adding up all of the magnetic moments of all of the atoms leads to a very strong permanent magnet.

In commercial manufacturing, many strong magnets are made by a process called powder metallurgy. During this process, raw materials are melted in a furnace, and then cooled into some shape - sometimes in molds, sometimes in strips. These chunks are then crushed into very fine powders and compacted -- pressed together into blocks -- under a magnetic field so that all the particles in the powder will have magnetic fields that point in the same direction. After compaction, the blocks go into another furnace to be heated into solid masses. This step is called sintering and may shrink the magnetic blocks by 15-20%. Some magnets then require treatment at lower heat. Then, the sintered magnets may be ground smooth, cut into smaller chunks, or otherwise processed into specific shapes and sizes. They may also receive coatings to prevent rusting and reacting with chemicals.

The magnets are magnetized under another magnetic field, and some magnets, as a finishing step, will be stabilized to prevent the loss of their magnetization, and/or calibrated to achieve a specific strength of magnetization. Note that the process above describes how Samarium Cobalt and neodymium-iron-boron magnets are made. Other very strong magnets may be made through different processes. Source: magnet manufacturing process.

There are several ways magnets can be made. First, some materials are naturally “permanently magnetic.” This means the material will exert a magnetic force without any outside influence. Iron ore magnetite is an example of this. “Non-natural” permanent magnets can be made by subjecting special materials to a magnetic force. When the force is removed, these materials retain their magnetism. A final way to make a magnet is to make an electromagnet, rather than a permanent magnet. These magnets can be turned on and off with electricity. When electricity flows in a circular manner, it creates a magnetic field. So, by wrapping wires in a coiled fashion and turning the flow of electricity on/off, a controllable “electromagnetic “can be created.

The most powerful magnets are neutron stars which are collapsed stars. Stars have magnetic fields, and when they collapse to become neutron stars, the magnetic fields also contract. I do not understand stellar physics well enough to understand how neutron stars can exert magnetic fields, however.

The most powerful magnets made by humans are electromagnets: they use an electric current to create a magnetic field. Any moving electric charges will create a magnetic field.

Very strong magnets are made of elements like neodynium, not simply magnetite, which is an iron oxide* that occurs naturally but is not a particularly strong magnet.

From Arnold Magnetic Technologies:
There are several processes for making magnets, but the most common method is called Powder Metallurgy. ... Ferrite, Samarium Cobalt (SmCo) and neodymium-iron-boron (neo) magnets are all made by this method. Unlike ferrite, which is a ceramic material, all of the rare earth magnets are metal alloys.

From Wikipedia:
*Magnetite is a rock mineral and one of the main iron ores, with the chemical formula Fe₃O₄. It is one of the oxides of iron, and is ferromagnetic; it is attracted to a magnet and can be magnetized to become a permanent magnet itself. It is the most magnetic of all the naturally-occurring minerals on Earth.