The custom in many communities I have been to is for there to be a siyum on a mesechta on the day of taanit bechorot. All first borns who would otherwise have to fast attend the siyum and are thus exempt from their fast. I would like to know why that is so and why this practice is not utilized for other fast days.
R. Tzvi Reisman has a long, in-depth discussion on this topic here, and the following are some of the main points which answer the question.
1) The obligation of this fast is unlike other fasts, in that it is only mentioned in a Baraissa of Mesechta Soferim, and not the gemara. Thus the obligation to fast started only as a minhag and so it is treated more leniently than other fasts. In addition, the gemara Yerushalmi seems to imply that one should not fast on Erev Pesach.
2) Several poskim explain that due to the weakness of our generation, if the firstborns fast on Erev Pesach they will not be able to prepare for the Festival, and may be too weak to fulfill the mitzvos of the Seder. Therefore, since this fast is already more lenient than other fasts, using a siyum to exempt the fast is the better option than fasting.
3) Another reason that some poskim mention is that there is an implicit understanding that a person does not intend to fast if there is going to be a siyum on the day that he intends to fast.
Rabbi Elazar Mayer Teitz taught on 7 Nissan in 2008 that all other fast days include both an obligation to fast & an obligation not to eat. This means that even if one has eaten for whatever reason, one may not continue eating. Taanit bechorot includes only an obligation to fast. Once one is permitted to eat at a siyum or other seudat mitzvah, one no longer is fasting and may eat the balance of the day.
In addition to the standard answer provided by user4523, I heard an explanation that the custom dates to the time of the Temple, when everyone would visit to bring the pesach-offering. The firstborns would observe the Kohamin serve in the Temple, and realize that if not for the sin of the Golden Calf, it would be the firstborns doing the service. Thus the firstborns would feel depressed that they lost their opportunity, and would not eat.
However, the last mishnah in Horayos says that a Torah scholar is greater than a Kohen. By making a siyum, the firstborns comfort themselves by making themselves better than Kohanim. Thus consoled, they are able to eat.
According to this explanation, it is unclear if one can be exempt from the fast due to other types of se'udas mitzvah, such as a circumcision or pidyon haben.