The origin of this phrase:
The phrase "放鸽子(fàng gēzi)" mainly means that one doesn't keep his promise. The meaning of the word "放(fàng)" means sending something. And "鸽子(gēzi)" refers to a dove, a kind of bird. The phrase is from a story. In ancient times, people mailed letters by doves. One time, two people agreed to write letters to each other, but one of them sent a dove without letters. Then another people said: "why did you just send the dove? And you haven't kept your promise." So the phrase is used with the meaning that one hasn't kept his/her promises.
The use of the phrase:
The popular meaning of the phrase is used in the following situation: one makes an appointment with another, but the latter fails to keep the appointment. So the former is "被放鸽子(bèi fàng gēzi)." You can say "我被朋友放鸽子了(wŏ bèi péngyou fàng gēzi le)" my friend fails to keep the appointment to express that your friend didn't keep his/her words with you.