1) The day after Rosh Hashanah, 3 Tishrei is known as "Tzom Gedaliah - The Fast of Gedalia".

It is a fast day that was instituted by the Nevi'im (prophets) to commemorate the death of the prophet Gedaliah the son of Achikam, who was the leader of the Jewish people during the period after the destruction of the first Bais HaMikdash.

After his assassination at the hands of rival Jewish factions, thousands of Jews who still remained in Eretz Yisroel after the destruction of the Bais HaMikdash, were then driven out by Nebuchadnezzar HaRasha into Galus (exile), and thousands more were killed.

Eretz Yisroel remained desolate and barren, and thus this fast day was established for all of Klal Yisroel throughout the ages.

Some commentators (Radak Yirmiyah Perek 45, Rabbeinu Yerucham brought in the Bais Yosef Siman 549, and others) say that Gedaliah was actually killed on Rosh Hashanah, but as not to institute a fast on Rosh Hashanah, it was pushed off to the following day.

Many Poskim , therefore, are more lenient on this fast day regarding nursing and pregnant women, mildly ill people, fathers making a Bris etc., as they maintain that this fast day always has the status of a "fast that was pushed off". (See Mateh Ephraim Siman 602:3)

Others (including the Taz quoted in Biur Halacha Siman 549)  maintain that since it was originally instituted on this day (as well as the fact that many Meforshim ,based on the Talmud Rosh Hashanah 19a, and the Rambam Perek 2 of Hilchos Taanis, maintain that he was indeed killed on 3 Tishrei, and not on Rosh Hashanah), it has all the Halachos of the other fast days.

A Rav should be consulted to determine if any individual with any special situation may in fact be lenient on this fast.

2) The main point of this fast [and all fast days] is to bring our hearts to Teshuva. (See Rambam Hilchos Teshuva Perek 5:1 and Mishna Berura Siman 549:1).

It is good to increase the amount of Tzedakah one giveson a fast day. Many have the custom to calculate the amount that their meal(s) (that they aren't eating today) would have cost, and give that amount to poor people instead.

It is worthwhile to minimize working on Tom Gedalia, as well as during the entire Aseres Yemei Teshuva, to the best of one's ability, and to try and spend more time learning and davening in Shul and/or the Bais Hamedrash (See Chayei Adam Klal 143)