Caliber is a term given by the manufacturer for a watch movement.
Winding gives information on whether a movement sources power via a self-winding (automatic) or a manual winding mechanism. A self-winding mechanism generates power with a rotor whenever the watch is moving, for example while being worn. On the contrary, a manual winding movement has to be wound up regularly by hand.
In-house movement is the term for a movement that has been developed and manufactured by the watch manufacturer. Often manufacturers acquire their movements from third parties and focus on refining them. To develop and to produce an in-house movement requires much more resources and becomes more and more a trend for luxury watches.
Alternations/h describe the frequency at which the heart of a movement beats. With each alternation the balance wheel generates a push for the watch's hands. For example at 28,800 alternations per hour the watch's hands advance 8 times per second.
Chronometer is the term for a mechanical watch, if it's officially certified by the Swiss inspection body COSC to have an extraordinary accuracy. Each and single Chronometer has been tested by the COSC to receive such a certificate.
Complications are additional functions of a mechanical watch. Complications increase the complexity of a movement and normally come with a much higher effort in development and production. Among popular complication count:
date, second time zone, Chronograph, flyback function for Chronographs, rattrapante function for Chronographs, moon phase, indicator for power reserve, perpetual calendar, acoustic repetition of minutes or hours, tourbillion to compensate for the influence of gravitation.
Amount of jewels quantifies the number of jewels (in most cases synthetic rubies) that have been built into the movement. Jewels are used to reduce abrasion for moving parts in a watch's mechanical movement. In general, watches with a high degree of complexity do require more jewels.
Power reserve describes the period of time from the full winding of a mechanical watch until the watch stops without any additional winding.