Chinese New Year - 新年

How to say Chinese New Year in Mandarin?

春节 chūnjié 'Spring Festival' (used in China)

新年 xīnnián 'New Year'

农历新年 nónglì xīnnián 'Lunar New Year' (because it follows Lunar calendar)

How do you say/congratulate Happy New Year?
恭喜！恭喜！gōngxǐ gōngxǐ '(respectfully) wish (you) happiness'
恭喜发财！gōngxǐ fācái 'Wish happiness and prosperity'
新年好！xīnnián hǎo 'New Year Good'
新年快乐！xīnnián kuàile ' New Year Happy'

What can you wish people?祝你。。。zhūnǐ (wish you)... ＋
        身体健康 shēntǐ jiànkāng 'Good health'
        万事如意 wànshì rúyì 'Everything as you wish'        

Married People or Senior people will give 红包 hóngbāo (Red Envelope containing cash) as a symbol of luck to unmarried people, kids or lower-ranking people (like employers to employees).

Here is a popular song called Gongxi Gongxi with Pinyin that you will hear in every Chinese community during Chinese New Year:

Learn more:
http://laowaichinese.net/text-message-greetings-for-chinese-new-year.htm
http://mandarin.about.com/od/chineseculture/a/newyear.htm