Magnet
A magnet is an object that produces a magnetic field, attracting iron and other magnetic materials and having a north and a south pole.
Definition
A magnet is an object that pulls on iron, steel, nickel, and cobalt without touching them, with every magnet having two ends called poles, a north and a south; opposite poles attract and like poles repel, so two bar magnets brought together will either snap together or push apart depending on which ends face each other. It is a material or object that produces a magnetic field and exerts force on magnetic materials and other magnets; cutting a magnet in half produces two smaller magnets, each with both poles, since isolated poles do not exist, permanent magnets keep their magnetism because their atomic magnetic domains stay aligned, and Earth itself is a giant magnet, which is why a compass needle points north. Magnetism originates in moving charge and the intrinsic spin magnetic moments of electrons; in ferromagnetic materials exchange interactions align neighboring atomic moments within domains, and a permanent magnet is one whose domains have been aligned and pinned, losing that alignment above the Curie temperature ($770^\circ\text{C}$ for iron), while a bar magnet is well approximated by a magnetic dipole with moment $\vec{\mu}$, experiencing torque $\vec{\tau} = \vec{\mu} \times \vec{B}$ and energy $U = -\vec{\mu}\cdot\vec{B}$ in an external field, and Gauss's law for magnetism, $\nabla\cdot\vec{B} = 0$, reflects that magnetic monopoles have never been observed.
Example
A refrigerator magnet sticks to a steel door, and a compass needle, itself a small magnet, has its north-seeking end point toward Earth's geographic north, meaning that location is actually a magnetic south pole since opposites attract; touching an ordinary iron nail to a magnet turns the nail into a temporary magnet whose domains line up and then slowly scramble again once it is removed. A small neodymium magnet with dipole moment $\mu = 1.0\ \text{A}\cdot\text{m}^2$ placed perpendicular to Earth's field ($B \approx 5 \times 10^{-5}\ \text{T}$) feels a torque of $\tau = \mu B = 5 \times 10^{-5}\ \text{N}\cdot\text{m}$, enough to swing a lightweight needle into alignment, and rotating it from aligned to anti-aligned takes $\Delta U = 2\mu B = 1.0 \times 10^{-4}\ \text{J}$.
Key Insight
Every magnet is ultimately a collection of tiny current loops and spinning electrons, and breaking a magnet down to a single electron still leaves it with a north and south pole, which is why no one has ever isolated a lone magnetic pole.