מְגִלָּה - literally, scroll, often referring to the scroll from which the Book of Esther is read publicly. Root is ג.ל.ל (g.l.l) meaning rolling.
תַּעֲנִית - a fast, synonymous with צוֹם, related to the word עוֹנִי meaning poverty and strife
מִשְׁלוֹחַ מָנוֹת - literally, deliveries of portions, referring to the gifts of food Jews give one another on Purim. Term appears in the Book of Esther.
מַתָּנוֹת לָאֶבְיוֹנִים - gifts for the poor, which Jews give on Purim. אֶבְיוֹן is one of the many Biblical-Hebrew words for a poor person. Term appears in the Book of Esther.
סְעוּדָה - a meal, synonymous with the more widespread word in Modern Hebrew, אֲרוּחָה. Root is ס.ע.ד (s.a.d) meaning satiation.
מִשְׁתֶּה - a feast, a banquet, deriving from the root שׁ.ת.ה meaning drinking. In the Book of Esther, משתה figures prominently in the plotline.
פּוּרִים - plural of פּוּר meaning lottery, possibly from the ancient Semitic language, Akkadian
שִׁכּוֹר, לְהִשְׁתַּכֵּר- drunk, to get drunk, from the Biblical and Modern Hebrew word for liquor, שֵׁכָר
לְהִתְפַּכֵּחַ - to sober up. See last year's YDDH entry on this.
אָזְנֵי הָמָן - these triangular cookies resemble the ears of (the Purim villain) Haman, which is what the Hebrew term means literally
תַּחְפּוֹשֶׂת, לְהִתְחַפֵּשׂ - think of the children... all dressed up. Literally, a costume and to get dressed up.
לִתְלוֹת - to hang. Can be done with laundry or with naughty people such as Haman.
אִגֶּרֶת - a letter, as appearing in the Book of Esther. In Modern Hebrew, מִכְתָב refers to your garden-variety piece of post, while אגרת refers to letters written for more official occasions.
וְנַהֲפוֹךְ הוּא - and it was turned upside down, the classic term from the Book of Esther, embodying the spirit of Purim