This Chinese game is popular in Hong Kong and parts of Southeast Asia and is also played to some extent in the USA. It is known by several different names.

In Cantonese it is called Sap Sam Cheung (thirteen cards), which means 13 cards, and in Vietnamese it is known by the similar name Xập Xám Chướng .
In Chinese, it is also sometimes called Luosong Pai Jiu (罗宋Pai Jiu), which I think means Russian Pai Gow. The game is indeed distantly related to Pai Gow.
In the USA it is often known as Chinese Poker or sometimes Russian Poker, but note that some people also use the name Chinese Poker to refer to the climbing game Big Two. In Hawaii it is called Pepito.
In the Phillipines it is known as Pusoy, again not to be confused with Pusoy Dos, which is Big Two. Another name sometimes used is Good, Better, Best, referring to the three hands of a player.
The aim is to arrange your 13 cards into three poker hands - two of five cards and one of three cards - which will beat the corresponding poker hands made by the other players.

A recent development is Open Face Chinese Poker, in which after the first five cards, hands are built face up one card at a time.