Chinese Tea Custom

In casual occasions, there is tea; in serious occasions, there is, again, tea. The Chinese world would almost not function if tea is taken away from it. 
In marriages 
We describe parents who long for their sons to get married as "wanting to drink 'daughter-in-law's tea'". 
In traditional Chinese marriage ceremony, the bride and groom kneel in front of their parents and serve them tea. That is a gesture of gratefulness. It's like saying "thanks for bring us up. Now that we are getting married. We owe it all to you."
To show respect 
Younger generation greet elder generation with a cup of tea. That is a way to show their respect.
One note is that, in organizations and families, only people of lower rank serve tea to higher rank people. At least it was like that in the old days.
Today, the society is more liberal. Parents may pour kids a cup of tea at home, bosses may pour subordinates a cup of tea at restaurants. But it's just parents and bosses being nice. It would be inappropriate for low rank to expect high rank to serve tea in formal occasions.
Apology 
When we have a serious apology to make and words are not enough, we "pour tea and apologize". That is an act of regretfulness and submissiveness.
Say "thanks for tea" 
After guy A pours a cup of tea for guy B, you see guy B knocking his bended index and middle fingers (or similar varieties of finger tapping) on the table. You bet your savings that are secret agents. You are broke. They are just Chinese. 
The story goes like this. In the Ching Dynasty some 300-400 years ago, the emperor liked to dress casual and visit his kingdom. Servants were told to stay low profile in order not to reveal their master's identity. 
One day in a restaurant. The emperor, after pouring himself a cup of tea, filled the servant's cup as well. To the servant, it was a huge grace having the emperor pour him a cup of tea. Out of reflex, he wanted to kneel down and thank his master. But he was stopped because that would give away the emperor's identity. So instead of kneeling on his knees, the servant kneed with his fingers. 
That "thanks" knock is still in use today in the 21st century.