Light finger tapping is a custom for thanking the tea master or tea server for tea. After a person’s cup is filled, that person may knock their bent index and middle fingers (or some similar variety of finger tapping) on the table to express gratitude to the person who served the tea. Although this custom is common in southern Chinese culture such as the Cantonese, in other parts of China it is only acceptable if for some reason you cannot actually say thank you at that moment, for example if you are in the middle of talking with someone else at the table.
This custom is said to have originated in the Qing Dynasty when Emperor Qian Long would travel in disguise through the empire. Servants were told not to reveal their master’s identity. One day in a restaurant, the emperor, after pouring himself a cup of tea, filled a servant’s cup as well. To that servant it was a huge honour to have the emperor pour him a cup of tea. Out of reflex he wanted to kneel and express his thanks. He could not kneel and kowtow to the emperor since that would reveal the emperor’s identity so he bent his fingers on the table to express his gratitude and respect to the emperor.
The bent fingers for knocking are technically supposed to be three to signify a bowing servant. One is the head and the other two are the arms.
It should be noted that in formal tea ceremonies nodding of the head and/or saying “thank you” is more appropriate.