Varying Customs on Kosher

Being that Kosher law isn't one specific law, but rather a collection of thousands of years of laws, rabbinic dictums, and customs it can be quite complex. Additionally, Jews have different levels of religiosity,  come from different sects and communities, and hold different rabbinic opinions on Kosher law. This means that what Kosher means to one Jew may be quite different from what it does to another. When I say "up for debate" it means that either religious scholars debate it or that perhaps all religious scholars agree but in the modern world different people hold to different standards. I will explain some of the major points of divergence among different  Jews:

Certification/Supervision. Many Jews feel that certification is unnecessary. That so long as the ingredients are Kosher, it is satisfactory. This is where the idea of "eating out dairy/vegan/vegetarian" comes from. Though in their kitchen they may require all products to be certified or at least meat be so, they are fine eating food in other establishments provided they don't contain meat. However, many in the Orthodox community reject this idea, due to observance of some of the less-kept Kosher laws, customs, and stringencies explained below. They would require any food consumed to either be certified or prepared by a trusted person. As to what a certifier must do is also up for debate. This has led to many certification agencies with different levels of stringency and trust among the Orthodox community. Some may trust a certifier to certify carbonated water but not dairy products. 

Separation of Meat and Dairy. One of the most basic Kosher laws is the separation of Meat and Dairy products. The extent of separation depends on a person's background. Some Jews will have them so long as they aren't cooked together, meanwhile, others will have separate sinks for cleaning kitchen equipment used with Meat and Dairy products. Orthodox Jews generally prohibit any cross-contamination between meat and dairy products. This leads to establishments and factories being certified either "Kosher Meat" or "Kosher Dairy" (or "Kosher Parve" for neither).  Many Orthodox Jews will wait between consuming Meat and Dairy products for as long as 6 hours (although often a shorter period if the other way around). 

Equipment. Many Jews do not allow for cross-contamination between Kosher and non-Kosher as well as between Dairy and Meat. What to do with equipment that was previously used with non-Kosher and how to "Kasher" it is up for debate. As is using dairy equipment for meat and vise-vera. According to most Orthodox Jews, all equipment that comes into contact or may come into contact with food, and any ovens (unless the food is wrapped properly), must be Kosher and only used for the type of food being placed in it. 

Glatt (Meat). You may hear the term Glatt Kosher used to refer to a "higher level" of Kosher. In actuality it only refers to one additional step in the slaughtering process that is already widespread in the US (with the notable exception of Hebrew National products), but not as much in Israel. This is to check the lungs of the animal post-slaughter to ensure the animal was not ill at the time of slaughtering. Most of the Orthodox community requires meat to be Glatt (or Chalak which means the same). 

Cholov Yisroel (Milk). Jewish law requires milk to be from a Kosher animal. In some regions, milk from non-kosher animals is more common and thus can be confused or intentionally miss-labled. Thus many Jews require certification for milk. In the US, for something to be called milk, it is legally required to be cow's milk. The USDA regulates this heavily. Therefore many Jews rely on the USDA's assurances and do not require milk to be certified Kosher. Others do not and require dairy products to be certified as "Cholov Yisroel". 

Yoshon (Grain). Jewish law requires grain grown in Israel to be "old". This means it took root in the prior cycle, which resets every Passover. There is a debate among religious scholars about whether this is also required outside of Israel. In the US for much of the year, all grain is "old". However, as it gets further from Passover many Jews will only eat products certified as "Yoshon". 

Bishul Yisroel/Pas Yisroel (Jewish participation). Jewish law requires that a Jew be involved in the cooking process (Bishul Yisroel) for food to be considered Kosher, but this is only for food "fit for a king's banquet". What food meets that criteria and to what extent participation is needed is up for debate. To many, it merely means that a Kosher inspector lights the pilot lights and starts up the equipment at the beginning of production but for others that participation must be more involved. For bread, there may be different standards (Pas Yisroel) and many have the custom of being stricter on it between Rosh Hashana and Yom Kippur (the days of judgment).  

Produce Checking. Due to the high prevalence of bugs in produce, many Jews require produce to be inspected before they are considered Kosher. The level of this inspection and who does it is up for debate. Many Jews simply rely on a quick visual inspection and/or USDA standards. Others require more scrupulus checking or other bug removal procedures. 

G'vinat Akum (Cheese). Similar to milk, rennets (a necessary ingredient in "hard" cheesemaking) must come from a Kosher animal, and that animal must have been slaughtered in a Kosher fashion. Today most "hard" cheeses are made using micro-biological renets which avoids this issue. As to which micro-biological rennets are Kosher is up for debate (as some are made with non-Kosher mediums). Thus many Jews hold that cheese requires certification from a reputable certifier. 

Seperation of Fish. Many Jews have the custom to not consume meat and fish together. This was originaly to prevent the accidential consumption of fish bones. However, unlike meat and dairy, cross-contact or contamination is perfectly fine. Many Sephardic-Jews extend this to dairy products as well. 

Confusion. Just reading these articles you may find Kosher complex and confusing. So do Kosher consumers. Kosher law already includes some anti-confusion measures and many agencies require facilities under their perview to take steps to make Kosher less confusing. This is despite some of those mesures not being a requirement of Kosher law.