Yekum Purkan is the Aramaic prayer that is recited immediately following the Haftara every Shabbat morning. It is a prayer for the welfare of Torah scholars as well as laymen and other members of the congregation who engage in charitable activities. Although it specifically mentions the scholars of "Babylon," which obviously no longer exists today, most interpret the word "Babylon" as a euphemism for the entire Diaspora. Nevertheless, there are those who do indeed replace the word "Babylon" with "the Exile" or "Diaspora" though most authorities oppose such a change and encourage one to leave the text of the prayer as is.
Following the two paragraphs of Yekum Purkan, an additional Misheberach is recited in honor of all those who volunteer for the benefit of the community. One who prays alone omits the second Yekum Purkan, as it refers to "the congregation" which is obviously not relevant when one prays alone. In fact, some authorities rule that both paragraphs of Yekum Purkan should be omitted by one who prays alone. Sefardim generally do not recite Yekum Purkan as part of their liturgy although they do recite a prayer for the welfare of the congregation that resembles the Yekum Purkan liturgical style.
Yekum Purkan is not recited on a Yom Tov that falls out on a weekday. This is in order to allow for the already lengthy services to end that much earlier so that people can get home and prepare their Yom Tov meal. Indeed, it is reserved specifically for Shabbat, as it is primarily a prayer for those who study the Torah, which was given on Shabbat. It is also suggested that Yekum Purkan is only recited on Shabbat in order to recall that the Exilarch would be present in the synagogue on Shabbat mornings and reciting it then was considered to be a gesture of honor toward him.
Yekum Purkan was written in Aramaic, as that was the vernacular language in Babylon where it was composed, just after the canonization of the Talmud. In fact, some sources teach that Hebrew was all but forgotten in Babylon. Interestingly, however, Yekum Purkan is not found in the Babylonian siddurim of Rav Amram Gaon or Rav Saadia Gaon. It is first found only in the 11th century Machzor Vitri with minor variations in the text from that which is in use today.
Some have asked why Yekum Purkan is recited in Aramaic if we are taught that one who prays in Aramaic will not enjoy the benefit of having the angels assist in one’s prayers being accepted. This is because the angels are said to “deliver” our prayers to God, and even intercede on our behalf. However, Aramaic is a language that the angels don’t understand, and therefore they are unable to assist us. It is explained, however, that prayers recited together by a congregation are an exception to this rule. This is because God Himself is personally attentive to prayers recited by a congregation. As such, there is no disadvantage to reciting it in Aramaic.
Although one should not hold anything in one’s hands, except a siddur, when praying, it is permissible for the chazzan to hold the Torah for Yekum Purkan should it be the congregation's custom to do so. In fact, it is quite appropriate to hold the Torah when reciting Yekum Purkan considering that it is a prayer for Torah scholars. In some communities, the one who leads the Shacharit service is the one who recites Yekum Purkan. In most congregations, however, it is the one who leads the Mussaf service that recites it.
For a fascinating and exhaustive treatment of Yekum Purkan, see: Rivevot Ephraim 6:455.
 Rema, OC 284:6.
 Mishna Berura 101:19; Rivevot Ephraim 1:216:1.
 Siddur Harashban 20b; Rokeiach 53.
 Ziv Hashabbat p.183. See Sefer Chassidim 1170 for another reason it is recited specifically on Shabbat
 See: http://en.wikipedia.org/wiki/Exilarch.
 Sefer Yuchsin p.121; Safra Chadeta p. 91 cited in Rite and Reason p. 252.
 Siddur Otzar Yisrael vol. 1 p. 704; Siddur Rokeiach p. 441, 561.
 Torat Moshe, Vayigash. Source provided by R' Yonatan Pachas.
 Sota 33a.
 Sota 33a; Sefer Kushiot 8. See also Har Tzvi, OC 1:64; Yabia Omer 1:33; Yechaveh Daat 3:43.
 Taz, OC 96:1; Mishna Berura 96:2; Minhag Yisrael Torah, OC 286:3.
 Rivevot Ephraim 4:97:1.