Here is a good explanation from Chabad on the problem of Agunah:
http://www.chabad.org/library/article_cdo/aid/613084/jewish/The-Agunah.htmHistorical Overview of the Agunah IssueAgunah means "anchored," or "chained." An agunah is a married woman who is not living with her husband, but has not been released from the bonds of matrimony. Though she wishes to put her marriage behind her, she is not free to remarry. She is chained to an unwanted marriage.
According to halachah, a woman may not remarry unless there is clear evidence that her husband had died or halachically divorced her with a get document. In times past, most agunot (pl. form of agunah) were victims of vanishing husbands. Traveling businessmen were often killed by bandits, who would dispose of the body leaving behind no trace of evidence. Or, a traveler would die in a remote location, and due to the lack of communication technology, or proper identification on the husband's person, the wife would remain uniformed. Frequent pogroms and wars habitually left agunot in their wake. Before recent times it was also fairly easy for an individual who had fallen upon hard times – or was unhappy with his current job, lifestyle, marriage, reputation, etc. – to simply vanish in the night and resurface in another city or country and start anew, unencumbered by previous obligations—including marital ones.1
The "agunah problem" was always one of the greatest challenges facing halachic authorities. The sages of the Talmud,2 recognizing the tremendous personal tragedy of the agunah, instituted various halachic leniencies intended to decrease the incidence of women in this woeful state.3 In biblical times, soldiers who were dispatched to the battlefield were required to first deliver a divorce to their wives,4 allowing the wives to remarry in the event that the husbands did not return.5 Halachic literature of the last millennium is dominated by "agunah responsa," penned by halachic authorities in response to inquiries from agunot who presented (sometimes flimsy) evidence of their husband's demise, and now wished rabbinical authorization to remarry. A precursory glance at these responsa reveals how these rabbis went to great lengths to find halachic precedent to declare these women widows.6
The Modern-Day AgunahToday the world is "smaller," and it is it is very uncommon for people to simply vanish. Nevertheless, the agunah problem persists, primarily due to husbands who cruelly refuse to grant their wives a divorce—despite rabbinical courts' orders to do so. A variety of reasons motivate these recalcitrant men. Many of them are unhappy with the financial aspect of their divorce settlement, others with custody arrangements, and they use the get as leverage in negotiations. Others hold their wives ransom, refusing to give a get until the wife pays an outrageous sum of money. While others refuse to give a get simply out of malice and spite.
Ironically, this type of agunah, the one whose husband is very much present but refuses to give a get, is a relatively new phenomenon. According to halachah, though it is the husband who gives his wife the get, a woman too may demand a divorce if she can prove that the husband is neglectful, repulsive or abusive. In such an instance, the halachah is unequivocal:
"One who is halachically required to divorce his wife and refuses to do so, a Jewish beth din – at any place and at any time7 – corporally punishes him until he says, 'I wish [to divorce].' The get is then written and it is a kosher get."8In short: the beth din is empowered to use any and all methods at their disposal to compel the husband to "agree" to divorce his wife. This includes imposing sanctions on having casual or business dealings with the noncompliant husband, and even using brute force if necessary.9
These measures were enough to induce the vast majority of people to comply with the rabbinical courts' decisions in these matters. This remained the case until relatively recently; because for the most part Jewish communities in the Diaspora were authorized to police, adjudicate internal disputes and dispense justice within their own communities.
Today, however, religious courts are not empowered to take justice into their own hands. Add to that the fact that today most Jewish communities are not united under the auspices of one specific communal body or rabbinical court, thus severely limiting the efficacy of any social or religious sanctions imposed by a given beth din.
Thus the agunah problem has evolved. The problem of vanishing husbands has practically vanished. In its stead we are faced with a new scourge: the recalcitrant husband.