First in a set — one component per paper. The toroid earns the opening slot because it isn't just a component, it's the birthplace of the whole field: the apparatus electromagnetic induction was discovered on was a ring exactly like this one. What it does in a circuit: store energy in a magnetic field and fight any change in current — the one component whose entire personality is resisting change.
A coil of wire wound on a ring-shaped core (usually ferrite or powdered iron). Electrically it's an inductor; wind a second coil on the same ring and it's a transformer. The ring shape is the point: it gives the magnetic field a closed loop to live in, so the flux stays inside the core instead of leaking out — which means high efficiency and very little radiated interference. That's why toroids show up wherever leakage would be a problem: power supplies, EMI filters, audio transformers.
Michael Faraday, 29 August 1831, at the Royal Institution in London. Faraday wound two separate coils onto opposite sides of an iron ring — his "induction ring" — connected one to a battery and the other to a galvanometer, and noticed that at the instant he closed or opened the battery circuit, the needle on the other coil kicked. Not while current flowed — only when it changed. That experiment discovered electromagnetic induction, and the device was, physically, a toroid. It was also the first transformer. So this component isn't just old — the ring is the exact geometry on which the relationship between electricity and magnetism was first caught in the act. Joseph Henry independently found self-induction around the same time (1831–32); the unit of inductance, the henry (H), is his.
Current in the winding makes a magnetic field (Ampère); the ring confines it. Then Faraday's rule does the rest: a changing field induces a voltage that opposes the change. Drag the slider to change the current and watch the induced voltage respond — steady current gives zero volts; fast change gives a big spike. The inductor is blind to current itself and reacts only to its rate of change.
The joule thief (the previous walkthrough) is the toroid showing off: interrupt ~50 mA through ~100 µH in about a microsecond and V = L·di/dt ≈ 5 V — a spike several times the 1.5 V battery, enough to light a blue LED the cell could never drive directly. Same trick, made continuous, is a boost converter in every phone charger. Wind two windings and reverse one and you have a common-mode choke — an EMI filter that passes your signal but blocks noise riding on both wires, because the ring's closed flux path cancels it. Every one of those is the same fact: a collapsing field makes a voltage, and a closed ring makes the field efficient.