The Ten Martyrs (Aseret Harugei Malchut עשרת הרוגי מלכות) refers to a group of ten rabbis living during the era of the Mishnah who were martyred by the Romans in the period after the destruction of the second Temple. Although all ten could not have been killed at the same time since two of the rabbis listed lived well before the other eight - they are listed together, in a manner of a dramatic poem (known as the Eleh Ezkera) recited on two important Jewish holidays, to elicit the proper mood of the day, one of reflection and the hope of redemption in the face of attacks to the beliefs of Judaism.
The term "martyrology" is also used about the story of the deaths (martyrdom) of several famous Rabbis (including Rabbi Akiva) by Romans, read both on Yom Kippur and Tisha b'Av. The deaths were gruesome, including, being wrapped in Torah scrolls and then being set aflame. See also Midrash Eleh Ezkerah.
The rabbis mentioned lived over a period of several hundred years and their stories are presented as a plot by Romans and others to weaken Jews by destroying Jewish leadership.
In the story, the Roman emperor Hadrian decides to martyr 10 rabbis as 'punishment' for the 10 brothers listed in the Torah who sold their brother Joseph to Ancient Egypt (Genesis 37). He justifies this by saying that the penalty for this was death (according to Jewish law, one who kidnaps his fellow Jew and sells him into slavery is punished with death. This, however, does not allow for descendants to be punished in place of their ancestors), and though this crime took place almost two thousand years earlier, there are 'none like you' 10 who are capable of rectifying this crime.
According to the poem, the first two to be executed were Rabban Shimon ben Gamliel and Rabbi Yishmael the Kohen Gadol. Rabban Shimon Ben Gamliel was beheaded, and while Rabbi Yishmael wept, the Roman ruler's daughter coveted Rabbi Yishmael for his physical beauty. When she was told that he would have to be executed as well, she asked that the skin of his head be flayed while he was alive, so she could stuff the skin and look at his face.
The most well known martyr is Rabbi Akiva, who was raked over his skin with iron combs. Despite the pain consuming him, he was still able to proclaim God's providence in the world by reciting the Shema, drawing out the final Echad - "One".
Another sage martyred was Rabbi Haninah ben Teradion, who was wrapped in a Torah scroll and burned alive. Damp wool was packed into his chest to ensure he would not die quickly. When he was being burnt, he told his students that he could see the letters of the sacred torah "flying up" to heaven.
The others mentioned in the poem are Rabbi Hutzpit the Interpreter (so named, because he would interpret the words of the Rosh Yeshiva - the head of the Yeshiva - for the masses, who could not follow all his words); Rabbi Elazar ben Shamua; Rabbi Hanina ben Hakinai; Rabbi Yesheivav the Scribe; Rabbi Judah ben Dama; and Rabbi Judah ben Baba.
Part of the Yom Kippur services
This poem is best known as part of the Yom Kippur mussaf recital in the Ashkenazi ritual. This was made part of these services because of the impact losing so many pillars of Judaism would have to the masses. As such, it has become one of the 'highlights' of the day, marking a point when the congregation should reflect on their own lives and the sacrifices that were made for their sake. A similar poem Arzei haLevanon is recited as one of the Kinot on Tisha B'Av.
In contemporary times
In contemporary times, the moral of this poem has taken on a new meaning with the deaths of millions of Jews during the Holocaust. Many Jews followed Rabbi Akiva's example reciting the Shema as they were being led to the gas chambers. A liturgical link was made explicit in the Mahzor for Rosh Hashanah and Yom Kippur, a 1972 project of the Rabbinical Assembly which is the primary rabbinical association for Conservative Judaism. In an elaborate reworking of the traditional text, the martyrology was interwoven with material from Hayyim Nahman Bialik, Hillel Bavli, and other sources, connecting the Roman persecutions to later persecutions such as those by the Russian Tsars and the Nazis. The section climaxes with a special version of Mourner's Kaddish which names sites of persecution and Jewish flourishing.