Agglutinative languages are somewhat harder to understand than other categories. it's easy to see what the difference between synthetic fusional languages (like Latin or Russian) to isolating ones (such as Chinese or English): in isolating languages you only have words mixed with each other in various ways, but no morphology (or at least not very much of it, since no language is pure).
But why shouldn't we count agglutinative languages as isolating? If they just add suffixes (or prefixes, to be inclusive: Bantu languages such as Swahili and Zulu use prefixes more than suffixes for indicating things). You can treat every suffix as yet another word and then they're all just a few words bunched together, no different than in Chinese.
But the truth is that agglutinative particles aren't words. Unlike "pre"-positions, they can't stand alone and they can't put themselves wantonly in both sides of words ("understand" vs "stand under") or entirely removed from the word they describe ("Which city do you live in?"). In fact, agglutinative particles are so tied to the "full" words they are attached to that they really are, for all intents and purposes, affixes. In most agglutinative languages, they are indeed written as prefixes and suffixes (i.e. there's no space between them and the words they describe) but in Japanese romanization (especially when targeted toward foreigners) they are not. This is mostly due to a westernized perception of Japanese in my opinion.
For the record, the modern language most similar to Korean, which is given as an example for slot, is most probably Japanese. They both have a verb conjugation strategy which can be explained through slots (though it's arguably simpler in Japanese than in Korean), but you won't find the word "slots" used in textbooks in neither of these languages (at least not the ones I've read).
Actually, "slots" is just a more disciplined way to view what is essentially a standard order for suffixes. In Japanese, as well as in Korean, certain things have to come before others. If your verb specifies negation for instance, it should definitely come after volition (～たい), but also before the past tense marker (～た). If you fulfill every possible category you can also count slots in Japanese, but it's less useful than in Korean, since in Korean some suffixes that are not attached directly to each other still need to come together in some of the cases, so it might make talking about slots more helpful, even for students (but it isn't - most Korean textbooks I've read are quite bad at making things simple :/)
Deep inside, Korean uses verbal bases, just like Japanese, and each suffix is attached to one of these verbal bases and can be conjugated to any of them (so the next suffix could be attached to it). The same thing can be said about Japanese, where the auxiliary verb だす is attached to the I-base of verbs (e.g. kaki for kaku) and the auxiliary verb あげる can only be attached to the TE-form. They both have further bases of their own, of course. Some other suffixes (such as ～ない and ～たい) behave more like i-adjectives, but as we've seen before, i-adjectives are, in fact, verbs.
Case inflections on noun are actually quite rare. As far as I know, the only major (or perhaps we should say: commonly mentioned) agglutinative languages to ever have cases are Finnish and Estonian. Some languages, such as Turkish, Hungarian or even Japanese are claimed to have cases, since there are particles that mark things like direct object (accusative) or the instrument of a verb (instrumental). These should only be called cases, however, if they are used for noun agreement, especially with adjectives agreeing to their nouns. In Latin they do (compare De Novo Mundo about the new world to Novi Mundi of the new world), and in Finnish they also do. Japanese is obviously different, but the same can be said for most other agglutinative languages.
In fact, there is probably no single grammatical category (be it case, gender, tense, person, number, volition, negation or whatever) that is done through agglutinative suffixes in all agglutinative languages. Being an agglutinative language has nothing to do with what is expressed through agglutination, but rather with how it is expressed.