Quick Answer:Magnets stick to iron, nickel, cobalt, and a few other materials because the atoms inside those materials can act like tiny magnets — and a nearby magnet's field can line them up. Materials like wood, glass, and plastic lack that atomic-scale magnetism, so everyday magnets ignore them.

The science behind magnetism

Magnetism is a force that can attract or repel objects from a distance, transmitted through a magnetic field. Fields arise from electric charge in motion — in a wire carrying current, or in the electrons inside a material. A field has a direction: it points the way a compass needle would, from a magnet's north pole toward its south pole. (HyperPhysics)

Magnetic materials

Iron, nickel, and cobalt — and alloys that contain them — are called ferromagnetic. In these materials, the magnetic fields of individual atoms can line up with one another in patches called domains, giving the material its own local magnetic pulls. Bring a paperclip near a magnet and the magnet's field lines up those atomic magnets with itself; the aligned pull adds up, and the paperclip jumps. (National MagLab)

The alignment is also why magnets can lose their magnetism: heating a magnet, or banging it hard, scrambles the aligned domains, and the atomic magnets no longer pull together. (LibreTexts Physics)

Why other materials don't respond

In most materials, atomic-scale magnetism either doesn't exist or points in random directions, cancelling out on average — so the material feels no noticeable pull from a nearby magnet. That covers wood, glass, plastic, water, and even most metals (copper and aluminum, for instance, barely respond). They aren't perfectly "nonmagnetic": all materials respond at least weakly, some very slightly attracted and others very slightly repelled, but the effects are so faint that everyday magnets don't move them. (LibreTexts Physics)

Poles and fields

A magnet's pull is strongest at its poles — which is why a paperclip sticks to the end of a bar magnet but not at its middle. Poles come in pairs: opposite poles (north–south) attract, like poles (north–north) repel. And cut a magnet in half as many times as you like: each piece still has both poles.

Heat, treatment, and magnetism

Two things can undo a magnet's alignment: heat and rough handling. Heating makes the atoms jiggle faster and can scramble the aligned domains; past a material-specific temperature (called the Curie point), the magnetic ordering breaks down entirely, and the metal stops being magnetic until it cools again in a field. Strong external fields can also re-magnetize a material in a new direction — which is exactly how data is written onto magnetic storage. (National MagLab)

Applications of magnetism

In everyday life

Magnetism is at work in daily life even when we don't notice it: refrigerator magnets, magnetic clasps on bags and jewelry, and the magnetic strips that store data on the back of a credit card.

In technology and industry

Hard drives store data magnetically; speakers and headphones turn electrical signals into sound with magnets; electric motors — from kitchen appliances to cars — run on magnetic forces. In industry, magnets sort magnetic materials from non-magnetic ones at recycling plants and lift heavy loads in scrapyards. Medical imaging relies on magnetism too: MRI machines use strong magnetic fields to map the inside of the body.

In scientific research

Magnetism remains an active research frontier. Scientists use it to study material properties at the atomic scale, and particle accelerators use magnets to steer beams of particles along their paths.

The bottom line

A magnet sticks to some things because those materials contain atoms whose tiny magnetic fields can be lined up with the magnet's own — and it ignores others because their atoms can't cooperate that way. It is a property of the material's structure and temperature, not of "metal" in general.

Sources & further reading

That’s the mechanism. The wonder is still allowed.