The strength of a magnet of a given material depends on size and shape, so a large iron magnet would be better at attracting a particular piece of metal compared to a smaller magnet.
The magnetism arises since the electrons in the iron all point the same way and the force between these electrons is what we call magnetism. The electrons in the other magnet are also lined up and there is an attractive interaction between all these electrons that brings the magnets together. If you have a big magnet, it has more of these electrons so its strength is bigger.
However, the strength of magnetism also depends on the material. Iron nickel cobalt are standard magnetic materials, but things like neodynium (a rare earth) are much stronger so need less material than iron would.
You can make non-magnetic materials magnetic by adding in iron or nickel to their structure and this creates small regions that are magnetic – make enough of them and your non-magnet turns into a magnet.
If you heat up a magnet you will see it magnetism disappears and upon cooling it comes back. Heating the magnet will make the electrons in iron point in random ways, so you lose the magnetism, cooling allows the electrons to line up again.
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