Aside from actually refraining from food, fast days that are established – either for an individual or for the community – are days of prayer and introspection. Extra petitionary blessings are added to the Amida prayer. On fast days connected with a lack of rain in Israel, for example, 24 blessings are recited (see dapim 15–16). Our Gemara asks whether an individual who accepts a personal fast will add an extra blessing to the Amida, or will simply add a prayer within the “catch-all” blessing of shome’a tefilla, in which we ask God to accept our prayers. (Our tradition today has individuals including a prayer for the fast day within the shome’a tefilla blessing, while the shaliah zibbur, who represents the congregation in his repetition of the Amida, says it as a separate blessing between the berakhot of go’el [redemption] and rofeh [healing].)
To respond to this question, the Gemara quotes a baraita, which rules that on an individual fast day the person will say an Amida of 18 blessings, while on community fast days, 19 blessings are recited.
Aside from the concern of the Gemara with the additional blessing for the fast, use of the term Shemoneh Esreh, i.e. 18, when referring to the Amida is, itself, interesting. Although we traditionally call the Amida prayer by that name – Shemoneh Esreh – a quick check of any prayerbook shows that there are 19 blessings in the standard Amida. Tosafot R”id discusses at length the common name Shemoneh Esreh, pointing out that the additional blessing of birkat ha-minin was added in Yavne, making it a long-standing part of the Amida.
The explanation for this anomaly seems to be based on differences between the Talmud Bavli and Talmud Yerushalmi (and, apparently, the tradition in Israel as opposed to the Diaspora). In Israel the two blessings before shome’ah tefilla -which deal with the rebuilding of Jerusalem and the reestablishment of the Davidic monarchy – were combined into one berakha. In that way even after the addition of birkat ha-minim there still were only 18 blessings in the standard Amida. In Bavel these two berakhot were recited separately and 19 berakhot were said in every prayer. Nevertheless, the prayer was still called Shemoneh Esreh based on the original count of blessings.