The balance spring is the regulating heart of any mechanical movement: without it, the balance wheel would oscillate without rhythm and the watch would measure nothing. I call it a hairspring when I want to be precise, because that's exactly what it is: an extremely thin ribbon wound in a flat spiral, anchored internally to the collet and externally to the spring stud. The oscillation frequency depends on its rigidity, its mass, and its active length. Modifying any of these parameters means shifting the regulation.
The historical problem with the hairspring is sensitivity: to mechanical wear, temperature variations, magnetic fields. Metallurgy responded first. Glucydur and then Nivarox reduced thermal drift. Nivachron — a niobium and titanium alloy developed by ETA — added resistance to magnetic fields without sacrificing the elastic characteristics of steel. You find it today in ETA Powermatic 80 movements and in Rado calibers like the R766. The practical result is a movement that doesn't lose seconds if you bring it close to a speaker or magnetic clasp.
Silicon changed the rules of the game in the 2000s. Rolex introduced the Parachrom — their own alloy, not pure silicon — and then the Syloxi, in paramagnetic silicon, mounted on the 2232 calibers and movements of the 1908 family. Breguet works flat silicon. Jaquet Droz and H. Moser use silicon hairsprings with double overlaid laminae: the so-called double hairspring, which improves isochronism and reduces distortion from friction between adjacent coils. Silicon requires no lubrication, is paramagnetic, and can be laser-cut with tolerances impossible for traditional metals. The downside I know from direct experience: it's fragile. A sharp impact on the silicon hairspring can break it; the Nivarox one bends and often recovers.
The geometry of the terminal curve is a chapter unto itself. The terminal curve — called Phillips terminal curve or Breguet curve depending on the profile — modifies the external closure of the hairspring to make it more isochronous: oscillation remains constant regardless of amplitude. Without this correction, the watch runs differently depending on the state of charge. Moritz Grossmann uses a curve calibrated according to Gustav Gerstenberger's designs; Chopard declares the Phillips terminal curve on the LUC caliber. These are details that make a real difference on the chronometer.
The mainspring — the one inside the barrel — is a different component, often confused with the hairspring. The barrel is the container that stores energy; the hairspring is the oscillator that governs it. Between the two is the train, the escapement, the escape wheel. When Dietrich Gruen patented the Safety Pinion in 1874, he was protecting the train from mainspring breakage: if the spring failed suddenly, the safety pinion would disengage before the energy destroyed the downstream mechanism.