There are three types of brakes used in railway operations: the Air Brake, the Steam Brake and the Vacuum Brake.
Vacuum Brake (or automatic vacuum brake): developed in the UK in the second half of the 19th century, the automatic vacuum brake was universally used in the UK and several other countries that adopted British technology, until air brakes became the standard in the late 20th century. Vacuum braking is still used on heritage railways in the UK and on some main line railway systems in Africa and India.
Dominic Wells has produced an excellent PowerPoint presentation that describes in great detail the operation of the train brakes, the first part of which focusses on Vacuum Brake operation. This first part (pages 1 to 205) can be downloaded by clicking here.
Air Brake: the most widely used brake system is the air brake which uses compressed air to generate the necessary force to apply the brakes throughout the length of a train. Air brakes are universally used on modern railway rolling stock worldwide. The second part of Dominic Well's presentation (pages 206 to 324) covers Air Brake operation (also in great detail), and this can be downloaded by clicking here.
If preferred, the combined file can be downloaded by clicking here.