The Shulchan Aruch (O.C. 170:1) writes (based on the Rambam, Hilchot Berachot 7:6; listen to audio for precise citation) that one may not speak during a meal, as this may cause food to enter his windpipe which could endanger his life. The Magen Avraham (commentary to the Shulchan Aruch by Rabbi Avraham Gombiner, Poland, 1637-1683), claims (in Se'if Katan 1) that this refers even to words of Torah. Since the concern here relates to the danger involved in speaking while eating, it is forbidden to speak even words of Torah.
The Mishna Berura (commentary to the Shulchan Aruch by Rabbi Yisrael Meir Kagan, the "Chafetz Chayim," Lithuania, 1839-1933) explains that this ruling of the Shulchan Aruch applies only during the time when one actually eats. In between courses, however, when one is not busy eating, it is permissible to speak. In fact, he emphasizes, there is a Mitzva for every person to speak words of Torah at every meal. The Sages admonish that people who conduct a meal without speaking any words of Torah are considered to have partaken from "Zivchei Meitim" (literally, "offerings of the dead"). The Mishna Berura cites the Shela Ha'kadosh (Rabbi Yeshaya Horowitz, Europe-Israel, 1565-1630) as requiring that one conduct at each meal a study session in either Mishna, Halacha, Midrash or Musar. The Shela emphasized that one does not fulfill this obligation of speaking words of Torah at a meal through the recitation of Birkat Ha'mazon at the end of the meal. At very least, the Mishna Berura adds, one should recite a chapter of Tehillim at every meal. He writes that it is particularly worthwhile to recite after the Beracha of "Ha'motzi" the chapter "Mizmor Le'David Hashem Ro'i" (chapter 23), which speaks of God's provision of livelihood, a most appropriate theme to discuss at a meal. It should be noted that many Ashkenazim have the practice to recite the chapter "Al Naharot Bavel" (Tehillim 137) before Birkat Ha'mazon on weekdays, and the chapter "Shir Ha'ma'alot Be'shuv Hashem Et Shivat Tziyon" (Tehillim 126) on Shabbat and festivals. This custom serves to ensure that one at very least recites a chapter of Tehillim at every meal.
The Shulchan Aruch Ha'Rav (Halachic code by Rabbi Shneur Zalman of Liadi, Russia, 1745-1813) comments (174:11) that it is proper for each participant in the meal to personally recite all Berachot, rather than allow one person to recite a Beracha on behalf of the others. For one thing, he writes, while eating one might not pay close attention to the Beracha, as is required when one seeks to fulfill his obligation by listening to somebody else's recitation. Secondly, as mentioned, speaking while eating poses a potential risk to one's life; hence, one would be unable to answer "Amen" to a Beracha he hears as he eats. Therefore, the Shulchan Aruch Ha'Rav rules, one should not recite a Beracha on behalf of others during a meal unless he first calls for their attention by announcing "Savri Maranan" ("Pay attention, gentlemen"). He adds that this is perhaps the origin of the common custom to announce "Savri Maranan" even before Kiddush on Shabbat and Yom Tov. In principle, this announcement is required only before reciting a Beracha during a meal, as discussed. However, so as to maintain a uniform practice and avoid confusion, it was established that one who recites a Beracha even before the meal should first call for everybody's attention by announcing "Savri Maranan."
Summary: While a person eats he must refrain from speaking even words of Torah. In between courses, however, it is permissible to speak, and everybody must make a point to speak some words of Torah at every meal, or at least recite a chapter of Tehillim.
