supersymmetry (SUSY)
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Nature's particles come in two great families with very different personalities. Matter particles, called fermions, are loners — no two can sit in exactly the same state, which is why atoms have structure and you cannot walk through a wall. Force particles, called bosons, are joiners — any number can pile into the same state, which is why a laser beam works. Supersymmetry is the bold idea that these two families are secretly two sides of one coin: that for every fermion there is a partner boson, and for every boson a partner fermion, related by a deep symmetry.
More precisely, a symmetry in physics is a change you can make that leaves the rules looking the same. Ordinary symmetries are things like rotating your experiment — the laws do not care which way it faces. Supersymmetry is a far stranger one: it is a rule-preserving swap between fermions and bosons. If it held exactly, every known particle would have an undiscovered 'superpartner' with the same charge and behaviour but the opposite family type, differing by half a unit of spin. The electron would have a partner boson called the selectron; the photon a partner fermion called the photino, and so on.
Supersymmetry is attractive for concrete reasons, not just elegance. The superpartners' quantum contributions would neatly cancel a troublesome over-sensitivity in the Higgs mass (the hierarchy problem), the three force strengths would meet at a single point at high energy hinting at grand unification, and the lightest superpartner could be the dark matter. The honest caveat: despite decades of searching, no superpartner has ever been seen. The Large Hadron Collider has pushed the simplest versions into a corner, so while SUSY is not dead, the easy, most-hoped-for forms are now strongly disfavoured.
In a supersymmetric world the electron (a fermion) is paired with a 'selectron' (a boson), and the quark with a 'squark'; experimenters hunt for these heavy partners by looking for collisions that produce a lot of missing energy carried off by an unseen lightest superpartner.
SUSY doubles the particle list — but no doubled partner has yet been found.
If supersymmetry were exact, the superpartners would weigh the same as their known partners and we would already have them; that they are not seen means SUSY, if real, must be broken, making the partners heavy.