Under the strictest rabbinical supervision
Contrary to popular misconception, rabbis do not "bless" food to make it kosher. Observant Jews do recite the blessings before and after eating, but these blessings cannot make the food kosher. Food is kosher if it meets the standards specified in the Torah, expounded in the Talmud and codified in the Jewish Law. There are three categories of kosher food - meat, dairy, and parve.
Meat. Any food made with meat, fowl or their derivatives, e.g., bones, broth or gravy, is called Fleishik / פליישיק - meaty in Yiddish. As was already mentioned, only meat of properly slaughtered, clean animals and fowl is kosher. The sciatic nerve (Genesis 32:33) and the fat / חֵלֶב, covering the kidneys and some other internal organs (Leviticus 7:23-25) may not be eaten and must be removed. All blood must be drained and major blood vessels cut out. The remaining blood must be removed either by broiling or soaking and salting the meat. This procedure is called koshering and it must be performed within 72 hours after slaughter, prior to cooking, freezing or grinding meet. It is required because of the biblical prohibition of eating blood (Leviticus 7:26-27, 17:10-14). Liver may only be koshered through broiling. A blood spot in an egg may be removed, but usually the whole egg is discarded. Reliable supervision is required at every step of processing meat to ensure its kosher status.
Dairy. Milk and milk products, such as butter, cheese, yogurt, etc., are called Milchik / מילכיק - milky in Yiddish. The source of the milk must be from a kosher animal. To ensure this, the milking process must be supervised and the resulting product is known as Cholov Yisroel / חלב ישראל. However, many rabbinic authorities maintain that in the USA and a few other countries, the government regulations that permit only cow’s milk to be sold commercially are stringent enough and may be relied upon in lieu of supervision.
Since hard cheeses are typically made with animal rennet (an enzyme used to harden cheese) and soft cheeses and other milk products may contain cultures and flavors that are not kosher, they all require supervision.
Parve. In Yiddish Parve / פאַרעוו means neutral - the term used to designate foods that are neither meat nor dairy. Eggs of clean birds, honey, kosher fish and its roe are kosher parve. Unprocessed fruits, vegetables, grains and legumes are also kosher pareve. However, care must be taken to check the produce for worms and insects, which are not kosher.
One must also ascertain that any of the products made from wheat, barley, oat, spelt and rye grown in (and according to a stricter opinion even outside of) the Land of Israel are made from Yoshon / ישן - old grain, i.e., grain harvested from the crops that have taken root prior to the annual Omer / עמר offering on the 16th day of the Hebrew month Nisan / ניסן.
In addition to this one must make sure that the fruits grown in the Land of Israel are not Orlah / ערלה - fruits from a tree under four years old that may not be used (Leviticus 19:23-19:24). Fruits grown outside of the Land of Israel are permitted, unless they are definitely known to be Orlah.
It is forbidden also to eat any produce grown by a Jew in the Land of Israel during the Shmittah / שמיטה year (Exodus 23:10-11). Plus, one has to make sure that various types of Terumah / תרומה and Maaser / מעשר - offerings and tithes described in Numbers 18:21-18:26; Leviticus 27:30-27:33, Deuteronomy 14:22-14:29, Deuteronomy 28:4, - are properly separated and Neta Rvai / נטע רבעי - the fruits of the four years old trees (Leviticus 19:24) - are redeemed.
Food mixtures. Pareve ingredients, e.g., vegetables, pasta, etc., are categorized as dairy, when cooked together with dairy food or as meat, when cooked with meat. Both mixtures are kosher.
However, a mixture of meat and dairy is not kosher. The biblical commandment not to “boil a kid in its mother's milk / לא תבשל גדי בחלב אמו” is repeated three times (Exodus 23:19, Exodus 34:26 and Deuteronomy 14:21). It is interpreted as a prohibition of cooking, eating uncooked or deriving any benefit from the mixture of meat and dairy. The rabbis extended this prohibition to the mixture of milk and poultry products as well.
There is a waiting period after eating meat or anything cooked with meat before eating dairy. No waiting is required after eating dairy, but one must wait between eating certain hard cheeses and meat. The prevalent practice is to wait six hours, but some wait only three hours and some have a custom to wait just a bit more than an hour. Rabbinical enactment prohibits eating fish and meat together, but no waiting period is mandated.
Kosher for Passover. There are additional dietary restrictions during Passover. No leavened products may be consumed on Passover. Therefore, certain foods made of grain products, although kosher the rest of the year, are not kosher for Passover. Plus, in many communities, legumes are not permitted on Passover.