A siderial watch measures siderial (star) time. A star (or other celestrical object) transits the upper part of a meridian every 23 hours, 56 minuites and four seconds, so a siderial day is shorter than a normal day measured by a normal watch. This was important to surveyors for the determination of longitude and azimuth.
The Society has two different Sidereal Watches in their collection.
Watch, Sidereal
Watch, Sidereal, Waltham