A Purity bond is a pledge that Bree was required to keep as part of her religion. The exact nature of the purity bond has never been revealed.
In the YouTube comments to the video Boy Problems... Bree first mentions that one of the reasons her parents would never allow her to date Daniel is their belief in purity bonds. Fans at the time, who would comb over the comments to every video, googled the phrase "purity bonds" and discovered that it did not appear to be used in any real religion.
Most people believed the purity bond to be something similar to a virginity pledge, where a young man or woman pledges not to have sex before marriage. Most signs pointed to Bree's religion having a connection to Aleister Crowley, but Crowley was known to be quite open about his sexuality.
In the video My Difficult Decision Bree says that her parents told her that she had violated her purity bond by sneaking out to meet Daniel and by kissing him.
The purity bond has not been mentioned since, and it is unclear if Bree still feels bound by it, whatever it was. However, in The Morning After, we see a half-naked Jonas in bed with a pink shirt hanging next to him. It is quite possible that as of this video, the purity bond (if it is indeed a virginity pledge) has been broken.
In the video The Bree Solution Taylor says that she thinks Bree may be trait negative as she broke the purity bond (hinting towards the night Bree spent with Jonas). However Spencer disproves that "breaking a purity bond" can change DNA. This video makes the theory that breaking the purity bond means losing one's virginity extremely possible as Spencer is more than likely talking about "sexual intercourse".
Understood in this sense, it is possible that the Order imposes a "purity bond" on ceremony girls to ensure that they do not contract a sexually transmitted disease that could then be passed to the Elders who receive their blood.