Kosher foods are those that do not violate Jewish dietary law. The law is complicated, and not all authorities agree on exactly what is kosher and what isn't. A kosher food will generally have on it somewhere a trademarked symbol belonging to a rabbinical supervising organization. This means the product has been investigated by someone belonging to that organization who is very familiar with Jewish dietary law, and has found that not only the ingredients but the process used does not violate any of the principles of that law as interpreted by that particular organization. Some organizations are stricter than others, but most of them make their guidelines for specific foods available so people know exactly what a kosher certification from that organization really means.
The question might arise: what might there be in a dill pickle that could possibly violate the dietary law to make some dills "Kosher Dills" and others just "Dill pickles"? The answer is that so-called "Kosher Dills" are not necessarily actually kosher in the sense of being made under rabbinical supervision, but that they're the traditional style (with lots of garlic and dill) that would typically be served in a New York kosher deli. The label usually actually reads "Kosher style Dill" with the word style written very small.
There is no dispute as to what is or isn't kosher. The only potential issue is that some people follow more strict standards. For example, cow milk in North America is considered kosher, however, some people will only drink milk that is 'chalav yisrael'. Chalav yisrael is a certification given to milk where the whole process of production is supervised by a 'mashgiach' (an orthodox observant Jew who is an expert on the laws of kashrut). A mashgiach can be a rabbi but that is not required. One of the only ingredients that could render a pickle not kosher is the type of vinegar used if a vinegar brine is used.