Translation:The special glasses are on the table.
Do you know if they're working at improving the speaking voice? She has some weird idiosyncrasies that make me miss questions every once in awhile. "Gläser" sounded like "Geleser" or something similar (i.e. she muffed the gl consonant cluster). "wird" often sounds like "will" or "willt" to me.
I'm not a native speaker, but the feeling I got is that "speziel" means more like "special purpose, specific" while "besonder" is "something different, extraordinary, something you cherish".
For example, "besonderer Freund" is a special friend whom you appreciate and like a lot, while "spezieller Freund" sounds weird to me and even a little dirty :D
So maybe "besondere Gläser" are made of cool material or were left to you by your grandma while "spezielle Gläser" are special glasses for wine only, not for water.
Again, this is just my feeling, I'm not a native
Does this refer to glasses that you wear, or glasses that you might drink out of?
'especial' is just an archaic alternative version of 'special' from the old-French habit of adding e- in front of Latin words that started with sp-, st- or sc-.
It's virtually non-existent in modern American and British English so I question its use here. You might come across it if you're reading an especially old book, but it's hardly a part of core vocabulary.
auf is a two way preposition sometimes takes Acusative If there is movement or direction but in this sentence takes dative because there's location
Ich stellte die Gläser auf den Tisch(Acussative) I put the glasses on the table
Die Gläser sind auf demTisch(Dative)
I hope that helps
Wiktionary is your friend. Just like English distinguishes between several types of raised surfaces for putting things on, so does German. A Tisch is a table. A desk (a table with cubbies/drawers for putting things inside and an exposed wall to fit a chair in) is a Schreibtisch. A nightstand (a small end-table that's put next to a bed) is a Nachttisch. A desert is a Nachtisch. In English you might call all of these things "a table" but you would probably get some pretty odd looks from people. You'd get a pass if it was something very close. But a counter, desk, and dining room table are all quite different things.