According to the Gemara, there are two reasons for washing mayim acharonim. Firstly, washing one’s hands removes the dangerous melach sedomis (Eruvin 17a and Chullin 105a). Secondly, one should wash one’s hands before reciting a beracha (Berachos 53b).
While Rambam (Brachos 6:2) and Shulchan Aruch (OC 181:1) both pasken that one should wash mayim acharonim, the Shulchan Aruch (181:10) brings Tosafos’s opinion that defends the practice of those who don’t, as we no longer need to be concerned about melach sedomis. The Rema (Darkei Moshe OC 181:1) writes that this is the normative askenazi custom. R’ Yaakov Emden (Mor Uketzia 181) explained that nowadays when people eat with cutlery, one doesn’t need to be so concerned about melach sedomis.
Nonetheless, the Mishna Berura (181:22) writes that the Vilna Gaon was particular to do so, and that ideally askenazim should wash. Although the Aruch Hashulchan (OC 181:4) writes that one doesn’t need to wash if one’s hands are clean and they wouldn’t normally wash them after a meal, he carries on to say that following the poskim, one should be careful to wash mayim acharonim and encourage one’s family to do so, too.
While R’ Ovadia Yosef (Yalkut Yosef 3:181:2) and R’ Ephraim Greenblatt (Rivevos Ephraim 1:140:3) hold that women are also obligated to wash, R’ Shmuel Wosner (Shevet Halevi OC 4:23) writes that mayim acharonim is a chumra that was never practiced by women (See Teshuvos Vehanhagos 1:174; Piskei Teshuvos 2:181:1).
Many are particular to cover up or remove the used mayim acharonim for Kabbalistic reasons (Kaf Hachaim OC 181:8). According to this view, one should specifically only use a few drops of water (Ben Ish Chai, Shelach 8). Interestingly, the Ben Ish Chai (ibid 7) writes that one should say the 3 words of this Halacha, Mayim Acharonim Chova, and thereby fulfil one’s obligation to learn at the meal, too.
The ashkenazi poskim don’t follow this view, however: The Aruch Hashulchan (181:7) and Mishna Berura (181:10) both write that one must use a revi’is (See Ma’ase Rav 84).Thus, while sefardim (who follow the Ben Ish Chai) should use the small mayim acharonim sets that are both small and hide the used water, ashkenazim who wash, should wash with a regular cup.