- You can say "农历"(formal name) or "阴历"(or even "夏历",which is less common) at the begining of the date to express that date is from the Chinese calendar.
- The name of the months:
正(zhēng)月(the first month),二月，三月，四月，五月，六月，七月，八月，九月，十月，十一月，十二月
Note: You could say "冬月" for "十一月" and "腊(là)月" for "十二月". Both of them are common to see.
- In the Chinese calendar,there is a concept of "leap month"(I wouldn't like to explain how it comes,because it's related to much Astronomy knowledge,hard for me to explain it in English),so one Chinese year may have 13 months,including an extra leap month. We use "闰(rùn)" before the name of a month when it's a leap month.I.e, this year,2012, the year of dragon, has a leap month, called "闰四月", so we have two "四月" this year: a normal "四月" and a "闰四月" after it.
Note: the chance of which month having a leap month is not uniform.I haven't met a leap "正月","十一月" or "十二月" (they are very very rare), but I think the corresponding leap months are "闰正月","闰冬月" and "闰腊月",based on some materials I read on this topic.
- For days, we say "初一","初二",...,"初九" for the first day, the second day,...,the ninth day; "十一","十二",...,"二十","二十一",...,"三十" for the eleventh day, the twelfth day,...,the twentieth day, the twenty-first day,...the thirtieth day. It's very easy if you can count number in Chinese.
Note: In calendar, you will see "廿(niàn)一", “廿二",...,"廿九", instead of "二十一","二十二",...,"二十九", but in speaking, I think you would prefer "二十一" to "廿一". "廿" is a classic character to refer to "二十".
Please see the image below to get a feel of the calendar commonly used in China, written in both Gregorian(Arabic numerals) and Chinese(chinese characters) calendar.
OK, too much stuff. Now it's the time to show some examples:
正月二十五(=正月廿五 in calendar)
Q: 你的农历生日是哪一天? What's your birthday in Chinese calendar
Q: 今年哪一天是四月十五? What's the day of "四月十五" this year?(here,the speaker wants to know what day in the Gregorian calendar is "四月十五" in Chinese calendar?)
A: 五月五号 the fifth, May
Q: 嘿，今年闰四月！你过哪一个生日？ Hey, this year has a leap "四月"! Which "四月十五" will you observe?