Sucking the milk from an animal:[1]

As a cure for heart pain: One who is moaning from heart pain of which his treatment is to suckle with his mouth from [the breast of] an animal, is permitted to suckle on Shabbos.

The Reason: As doing so is [merely] detaching in an irregular manner as it is not common to suckle milk [from the animal breast] with ones mouth but rather to milk it into a vessel and drink from it, therefore there is no Biblical prohibition involved in this nursing but rather merely Rabbinical and in a case of pain [the Sages] did not make their decree.

Due to hunger: However this allowance only applies to one with heart pain however one who is merely hungry it is forbidden for him to nurse from an animal on Shabbos.

However on Yom Tov it is permitted to nurse [from it] (if it is impossible to milk it through a gentile and also if he does not have food into which he can milk into).   

Summary-Sucking the milk from an animal:

One who is moaning from heart pain of which his treatment is to suckle with his mouth from an animal, is permitted to suckle on Shabbos. However one who is merely hungry is forbidden to nurse from an animal on Shabbos due to a Rabbinical detaching prohibition.

However on Yom Tov it is permitted to nurse from it (if it is impossible to milk it through a gentile and also if he does not have food into which he can milk into).