Answer: This is an article I once wrote on the subject:
Min HaTorah only milk that comes from a Kosher animal is itself Kosher. Therefore the animal from which the milk is derived must be from a Kosher species such as a cow or goat, or even a deer etc. Additionally the animal must not be a traifa. Since this is impossible to determine while the animal is alive, we rely on the majority of animals which are assumed to be healthy. In recent years this premise has been challenged based on the percent of traifos discovered in older milk cows, but this is outside the scope of this article.
By the times of the Mishna, see Avoda Zara 35b, Chazal were concerned that non-Jews may adulterate milk they sell with milk from non-Kosher species of animals which would be forbidden miD'Oraisa, and they forbade consumption of dairy products that were milked by a non-Jew without Jewish supervision. This milk is termed Cholov Akum and is assur miD'Rabannan. If this milk comes in contact with utensils while hot it renders them forbidden, just like any other non-Kosher food.
Since the logic behind the gezaira was based solely out of concern for possibility of consuming non-Kosher milk and not out of concern for assimilation and intermarriage, as we find with bishul and pas, there is a dispute among the early Achronim whether the issur applies in a location where no non-Kosher animals are found, or any other situation where no risk of contamination is present. The Pri Chadash and Radvaz rule that since there is no need to be concerned about dilution of the milk, direct Jewish supervision is unnecessary. However the Chochmas Adam and Chasam Sofer sharply disagree and rule that since Chazal created a gezaira to forbid Cholov Akum unless it was directly supervised by a Jew, it is a blanket prohibition applying even when the logic behind the takana may not be relevant. They rule that even under these circumstances the milking must have constant Jewish oversight.
There are different minhagim regarding butter made from Cholov Akum, which cannot be churned when containing milk from a non-Kosher species and is analogous to a situation where the concern that sparked the original gezaira does not apply. Those who are lenient consider it less problematic than the previous issue, because it is questionable if butter was ever included in the original enactment against milk and milk products since it is not mentioned in the Gemarra. Some Poskim also consider milk powder to be in the same category as butter, since it too cannot be made from non-Kosher milk. Shulchan Aruch YD 115:3 rules that one may follow a minhag to permit butter, but without a specific minhag to be lenient it should not be used without supervision. However, it is questionable if is these lenient opinions are relevant today, when modern advances in food science have enabled manufacturers to develop new methods and enzymes that could allow butter and powdered milk to be made from sources that were previously impossible. Also, many varieties of butter have liquid milk added to them after the churning stage to give them a richer, milky flavor. There is no assurance that this milk could not come from some different species of animal. Additionally the exemption for butter only applies if the milk was intended from the start to be used only for butter and the issur of Cholov Akum was never relevant to this milk. However, if the cow was milked for drinking or unspecified purposes, that milk is immediately forbidden and does not lose its Cholov Akum status to become Kosher when it is churned into butter.
Based on the concept of the Pri Chadash, mentioned above, HaRav Moshe Feinstein zatzal famously ruled in a number of teshuvos that any milk produced in a country that has laws forbidding adulteration of milk and has a framework for supervision of farms with significant consequences for those who violate the law, is considered to be permitted as if it were supervised by a Jew. Since the entire purpose of Jewish supervision is to insure that the milk contains nothing but milk from a Kosher animal, the government's supervisors can serve the same purpose. According to this opinion, even the Poskim who did not accept the logic of the Pri Chadash to preclude supervision would accept this virtual supervision to replace direct Jewish observation. However many Poskim disagree with Reb Moshe because they question whether the halacha is in accordance with the Pri Chadash, and they require direct Jewish supervision even when no significant risk is present. Additionally, some Poskim question whether the government supervision is rigorous enough to truly insure the purity and Kashrus of the milk. Reb Moshe himself writes in a number of places that a "ba'al nefesh" should be machmir and that he himself is careful only to drink true Cholov Yisroel. However the implication of his written words and the oral testimony of his family members is that he considered it a worthy chumra and not a halachic obligation. Each person should consult his personal Rav regarding how to conduct one's self.
Cheese is significantly more problematic than milk, because for the manufacture of cheese all Poskim agree that Chazal enacted a strict, absolute requirement for direct Jewish supervision since rennet from the stomach of a nevaila, an animal that was not slaughtered according to halacha, was typically used to congeal the milk into its solid state of cheese. Even when it is known with certainty that no problematic ingredients are present, it is universally accepted that a Jew must be present. In addition to watching the process, the Shach and other Achronim require a Jew to actively and physically participate in adding the enzymes to the milk. Most mehadrin hechsherim follow this opinion. This stringency only applies regarding the actual processing of the cheese, the halacha for the milk base is identical for cheese as it is for liquid milk, and one who relies on Reb Moshe's heter for Cholov Stam may use cheese derived from Cholov Stam, provided the cheese-making process was directly under Jewish inspection.