Although traditional Vietnamese culture has been influenced most by China, it is noticeably and uniquely Vietnamese. The many minorities (Khmer, Cham, diverse hill tribe peoples) have also had their influence, and there are plenty cultural and architectural traces of the French throughout the country. In addition, the American presence in the south has had a lasting effect. While rural areas and the older generations tend to be more traditional, modern "Western" culture, from food to shopping, is in full swing in the larger cities. This interesting mix makes Vietnam a fascinating country, and distinctly different from its neighbors.
The majority of Vietnamese consider themselves to be Buddhist. However, Confucianism and Taoism have influenced them in almost equal parts. Catholicism is also prominent, and many tourists are surprised by the number of Catholic churches in Vietnam, especially in the south. While near 10% of the population consider themselves to be Catholic, the older religions and philosophies of Buddhism, Taoism, and Confucianism still play a big part in their lives and customs. The latter three are seen in different aspects of Vietnamese life.
Confucian tenents have lead to respect for elders, they govern the way Vietnamese treat family, and have influenced the way they interact in social settings. Taoism, with its principles of balance and harmony, has affected everything from Vietnamese food to their worldview. And Buddhism provides the basis for their beliefs in reincarnation and karma, which also heavily influence day to day life.
Ancestor "worship" is also very common in Vietnam, with most Vietnamese having family altars. The ancestors are believed to help and protect the living, and are given small offerings of fruit and incense. To Westerners the idea of ancestor worship contradicts the Buddhist concept of reincarnation, but the Vietnamese are able to ignore this contradiction as a Westerner might ignore similar religious contradictions in their own beliefs.
The concept of "saving face" is another interesting aspect of Vietnamese culture, which is prominent in much of Asia. This concept can lead to both pleasant and frustrating situations for Westerners. Saving face may mean you'll be told a hotel has internet access when it does not, or that a driver knows the way to a particular location when he does not. Saying "no" is often seen as rude or incompetent, even if true! On the other hand, it makes Vietnamese far less likely to argue with you, and more likely to be helpful when you need it. Saving face also involves not speaking loudly or yelling, so you should avoid raising your voice in Vietnam.
Overall the Vietnamese people are socially conservative. You won't see people kissing in public, although holding hands is fine. To be respected and respectful of the Vietnamese, you should also restrain from strong public displays of affection. When entering a temple you should be modestly dressed (no shorts), and take off your shoes. Vietnamese people are very clean, and bringing the dirt on your shoes into a temple is considered very rude. Even on beaches, the Vietnamese tend to be well covered. Going topless is definitely out of the question.
For more information on Vietnamese culture, see our special on Vietnamese food.