1) Some people have the Minhag to fast on Erev Rosh Hashana. The primary reason for this fast is to bring a person to Teshuva.
Erev Rosh Hashana is the last day of the year, and Chazal tell us that one who does Teshuva on any one day throughout the year is considered as if he/she has done Teshuva for the entire year, therefore the Minhag is to rise early and fast on Erev Rosh Hashana and increase the amount of Selichos and Vidui etc., as this is the last chance to have a day of Teshuva in this year! (Chayei Adam)
It isn’t necessary to accept this fast upon yourself on the previous day (as is usually the case when one wants to fast on a day that isn’t a mandatory fast day) but one who is stringent and does in fact accept it upon him/herself is praiseworthy (Bach and Elya Rabbah)
2) Those who fast, should not do so for the entire day, as it isn’t good to enter Yom Tov hungry, rather this fasting is only necessary until after davening Mincha (and saying the prayer of Aneinu) or until Chatzos.
Today’s days, when the general public is a lot weaker [for sure spiritually] than they used to be, fasting on Erev Rosh Hashana is not prevalent. Many have a Minhag to instead partake in a Seudas Mitzvah such as a bris, Pidyon Haben, Siyum Mesecta and the like instead of fasting.
If one decides to indeed fast, it is best to stipulate [verbally or in their mind] that they are doing this "Bli Neder" (without a vow). If one didn’t make this stipulation, and had in mind to do this from now on every year, or if one in fact did this for three consecutive years, Hataras Nedarim (unbinding of pledges) would be necessary to be able to stop fasting on this day. ~~~~~~~~~~~~~~~~~~~~~~~~~~
Our holy sages, the Chazal tell us
תנא דבי אליהו כל השונה הלכות בכל יום מובטח לו שהוא בן עולם הבא, שנאמר הליכות עולם לו, אל תקרי הליכות אלא הלכות - one who studies [at least two] Halachos daily is assured a portion in Olam Haba - the world to come. (:מגילה כ"ח )