If electric current flows in a straight cable, it creates a magnetic field circling around it. If you bend the cable into a ring or a coil, then those little magnetic circles arrange to form electromagnet. What we take from this is that if electric charges are circling, there will be a magnet.

Now matte has lots of electron in it. Those things are electrically charged. But they also "spin". They are however infinitely small, so this spinning is a bit weird, but the same ideas apply. Hence they have a magnetic field by our previous observation; they are microscopic magnets. To get a macroscopic magnet, they need to be aligned so that their little magnets don't just cancel each other out.

In most matter, there are a lot of those little buggers, chaotically swirling around (metals, very hot "gas" [plasma]) or orbiting in atoms. The chaotic ones usually contribute nothing unless we create a current again or apply a magnet from the outside. They just point wherever they want.

Those in atoms follow some quantum rules, which tends to pair them in opposing directions; hence they cancel. They pair up, but sometimes one or more cannot find such a partner. This turns the entire atom into a slightly larger (but equally strong) magnet. Now most metals have those lonely electrons on the outside and they tend to join the swirling chaos, leaving their atom behind. But some atoms like iron or nickel have them a bit further on the inside where they stay. Thus those metals are effectively a lot of little magnets.

They are however still probably just pointing wherever they want. But if one gets them arranged properly in the same direction ("magnetizing"), then we finally got a magnet! This can be done in several ways, e.g. by an external magnet (might be electrical) forcing them to arrange; one usually then heats the piece of metal and cools it again, this will lock them in place so that they stay a magnet even if our external magnet is removed.