There are different kinds.
- The thermoluminescent dosimeter has a crystal that stores energy when exposed to radiation and then emits light when heated.
- An OSL (optically stimulated luminescence) dosimeter works in a similar manner, except that a laser is used to stimulate the light emission instead of heat.
- To measure neutron radiation, a commonly used dosimeter uses solid state nuclear track detectors (SSNTDs). For example, a small square of CR-39 (the same plastic used in many lenses for glasses) develops latent damage tracks as the neutrons collide with protons in the plastic. These latent tracks are later developed using a caustic solution, e.g., sodium hydroxide solution and the developed tracks are counted using a microscope.
Hopefully someone else can add how other types of dosimeters work by improving this answer for you.