Pour out your wrath upon the Gentiles that do not know you and upon the kingdoms that have not called upon your name. For they have devoured Yaacov (Jacob) and his shrine they have made desolate.
The above quotation is from the book of Tehillim (Psalms) chapter 79. Metsudat Dovid explains that in the context of the chapter, the word shrine is a reference to the Temple. (See also Rabbi Aryeh Kaplan’s, The Living Torah on Exodus 15:13 for further support for this translation of Tehillim/Psalms). This verse is also recited as part of the Haggada of Pesach (Passover) on Seder night.
The Ashkenazic custom at the Seder when we recite Blood, Fire, and Pillars of Smoke, the Ten Plagues and דצ”ד עד”ש באח”ב is that we dip our finger in our cup of wine and toss out a drop of wine for a total of 16 drops. The inspiration for this custom is to encourage the pouring out of Divine Wrath and Vengeance against the evil Gentiles. A modern popular explanation for the custom tries to do the exact opposite, namely, to discourage Divine Wrath and Vengeance.
Although this reason of “our diminishing joy due to the suffering of the Mitzrim” is widely quoted in many hagadahs today (and sometimes even wrongly attributed to the Abarbanel) it is actually a quite recent explanation that was first invented in the late 1800’s and then recorded in print for the first time by Eduard Baneth in 1904. This reason was then popularized among the masses when it was quoted in haggadahs that were printed in America in the 1940’s and 50’s. For example in a popular haggadah published in 1943 by the National Jewish Welfare Board “for members of the armed forces of the United States to address the compatibility of Jewish and American values” this is the reason given for the minhag.
The earliest reference to the minhag itself is actually found in the Pesach Drashos of the Rokeach (c. 1176-1238) and his mystical explanation of the sixteen drops of wine seems to be more in line with Rav Miller’s words above, in that it hints to the “sixteen sides sword of Hakodosh Boruch Hu.” The Sefer Maharil quotes the Rokeach and explains that we want the vengeance of Hashem “to fall upon our enemies.” The Darkei Moshe quotes the Maharil and writes that the minhag hints to the “angel in charge of vengeance.”