Beginning a Fast
The last meal that one eats prior to the start of Yom Kippur and Tisha B'av is known as the seudat hamafseket, meaning, “the separating meal.” It makes no difference whether one eats the seudat hamafseket moments before the fast begins, or even much earlier in the day. There is much discussion, however, as to when, and under what conditions, one is permitted to continue eating or drinking once one has eaten the seudat hamafseket.
According to some authorities, once one has made a mental declaration not to eat anything else for the remainder of the day, all further eating before the fast is forbidden. It seems that it does not matter whether one decided not to eat any more simply because one was full at the time or whether one actually intended to accept the fast upon oneself a little earlier than required. Other authorities, however, rule that eating only becomes forbidden once one specifically accepts the fast upon oneself. According to this view, all other considerations or declarations for not eating are meaningless (i.e. if one was full) and one may continue to eat right up until the beginning of the fast.
One who wishes to formally cease eating and begin the Yom Kippur or Tisha B’av fast, must do so verbally. This is due to a halachic principle which teaches that a mental declaration is inadmissible and has no halachic value. Other authorities, however, disagree and rule that even a mental declaration -- a promise or commitment made in one's heart – is indeed binding. As such, in deference to this minority opinion, one should exercise caution when mentally contemplating whether to perform a mitzva, such as when contemplating whether to donate money to a charity, or the like, as such thoughts may actually be legally binding.
There is an interesting difference, however, between accepting a private (voluntary) fast upon oneself and accepting a public one. When one accepts a private fast upon oneself, such as one who chooses to observe a fast over a disturbing dream or in honor of a Yartzeit, it is almost unanimous among halachic authorities that a mental declaration suffices for such a fast to take effect. However, as we have seen, when accepting a public fast upon oneself, the declaration to do so should be verbal.
Why the difference?
This apparent contradiction is actually resolved quite easily. As mentioned, when one contemplates the performance of a mitzva, only a verbal declaration is truly binding. One notable (and unanimous) exception to this rule is when one mentally commits to donate an offering to the Beit Hamikdash. Even a mere mental commitment to donate such an offering becomes binding upon a person to fulfill.
The distinction between a private fast and a public one is that when one accepts a private fast upon oneself it is the entire fast that is being accepted with one's declaration. Regarding a public fast, however, one's "acceptance" is merely an extension of an already existing obligation to fast (such as on Yom Kippur and Tisha B'av) which will automatically be binding shortly anyway, whether one formally accepts it or not.
As such, we see that accepting a private fast upon oneself is comparable to donating a voluntary offering to the Beit Hamikdash. Just like a mental declaration suffices for such an offering, it suffices for such a fast, as well. Extending an already existing fast, however, is not truly accepting a fast upon oneself. Rather, it is similar to accepting an extra mitzva, hiddur, or chumra upon oneself, for which a verbal declaration is required in order to formalize the acceptance.
 Rambam, Hilchot Taanit 3:3; OC 553:1.
 Rema, OC 553:1; Levush 553:1.
 Mishna Berura 553:2; Chayei Adam 134:8.