What are the primary methods used to detect ice on an airframe, and how do they work on at least a high level?
There are a few ways to sense ice on an airframe. The DOT has a document here that outlines the various types of sensors.
First and foremost planes have ice lights that project onto the wings so they can be visually inspected.
There are completely optical sensors that use a variety of methods to detect ice.
There are also some electro mechanical devices, here are a few examples, the rest are outlined in the DOT document.
Penny and Giles
The Penny and Giles system uses a probe type sensor that measures ice buildup in terms of Liquid Water Content (LWC). This system was designed specifically for use on helicopters and is available for all turbine-powered helicopters.
The sensing probe is driven magnetostrictively to vibrate at its resonant frequency of 40,000 Hz. As the ice detector enters an icing environment, ice collects on the sensing probe. The added mass of the accreted ice causes the frequency of the sensing probe to decrease in accordance with the laws of classical mechanics.
The Sunstrand system uses a mounted probe with a sensing surface facing the airstream. Beta particles are released from a Strontium 90 Radiation Source and are collimated by a specially shaped window in the probe housing. These particles pass across the sensing surface to a Geiger-Muller tube. High voltage is applied to the Geiger-Muller tube to enable it to detect the radiation emitted by the Strontium 90 source. Ice accretion on the probe decreases the beta particles detected by the tube. When the particle count drops below a pre-set level, a pulse rate discriminating amplifier energizes an icing signal visible to the flight crew. At this time a heater inside the probe is activated, the ice is removed from the probe, and the probe is ready to begin another icing cycle.
Some of these will work in a static situation while others may require the passage from a non icing environment to an icing situation.