I did not watch the drama, so feel free to correct any assumptions I incorrectly make, but I'm guessing that it might be referring to widows that did not re-marry after the death of their husbands.

Under the confucian paradigm that was introduced following the establishment of Joseon (in 1392) it was considered immoral for a woman to remarry on the death of her husband. As the years passed remarrieage was discouraged via both restrictions, in the form of loss of status for her and her children, and encouragements, in the form of being accorded a positive social standing. For reference, prior to the official enforcement of confucianism (i.e. during Koryo and earlier) there was no restriction on women re-marrying and typically they had equal inheritance rights.

Here's an extract from one of the discussions mentioned in the historical records about extending restrictions against women marrying thrice to women marrying twice. It's an example of the process that was involved in changing societal standards.

But now, according to the National Code [1469], thrice‑married women are listed together with licentious women, and their sons and grandsons are barred from the examinations and cannot receive posts in the censorial and administrative offices. Twice‑married women are not mentioned. Generally, statutes are based on fundamental law, and propriety is connected with human feelings. For a woman of a poor and lowly house who on neither side has supportive relatives, it is difficult to keep her chastity when she becomes widowed in early years. If her parents or relatives decide that she should marry for a second time, this does not harm propriety. ...

We think that the law of the National Code according to which the sons and grandsons of thrice‑married women do not receive high office should be strictly enforced and that Yi Sim’s wife, born Cho, should be severely punished. If we clearly point out what is good and what is bad, morality and mores will naturally become correct even if we do not make a new law concerning remarriage, and widows will understand this as a warning.

http://afe.easia.columbia.edu/ps/korea/neoconfucian_reform_women.pdf

Assuming that the drama takes place during late joseon and it involves a noblewoman (i.e. yangban) then it would've been completely forbidden for her to remarry. I recall reading about cases of women who widowed young and remained chaste for exceptionally long periods being mentioned in official documents for their upstanding decorum. I believe that this is the "chastity award" that you might have come across.

For commoner women a practice often carried out for remarriage was to "kidnap" the woman. The practice was known as "bossam" or "wrapping", where a man would wrap a widow and carry her to his home. From that point on they were effectively married. Sometimes it was carried out against the woman's will but most of the time it was with her prior consent. That way she avoided being a "loose" woman because she had been kidnapped against her will.