纸钱 (zhǐqián) has many different translations, including but not limited to, paper money, joss money and hell money. In order to understand why Chinese people burn paper money, you need to understand what they believe happens to the soul in the afterlife.
The Chinese believe in a place called Di Yu (地域 (dì yù), where all souls go before they are reincarnated. Your stay in Di Yu is not eternal, instead, you will remain in Di Yu for a period of time that the Ruler of Di Yu deems necessary in order for you to atone for your sins. Similarly, how comfortable your stay in Di Yu is will be also be determind by the ruler and his the 9 other Yama he presides over. And trust me when I say they have many inventive methods at their disposal to make you wish for a swift return to mortal realm!
Read A Journey Through Di Yu - "Chinese Hell" to learn more about Di Yu
So, during your not-so-wonderful stay in Di Yu, you will need money and material goods to get around, bribe officials and demons, to make your stay as comfortable as possible. And that money, as well as material goods, are sent to you by your decendents on earth through the burning of paper money in your name. Below are a few pictures of my mother and father-in-law - who were kind enough to agree to being photographed - burning paper money for their ancestors at Chinese New Year: