"Ta" indicates a particular thing, for situations where English would use "this," "that," or sometimes "the" (often just translated by itself as "this," but actual usage is a bit more complicated).
The English articles "a" and "the" (especially the former) don't generally translate into Polish at all. Their functions are left to context.
Also, while I'm shaky on the details of Polish language formalities, I understand "pani" to be more like "ma'am" than "woman." I'm not sure you would use it to talk about a woman in general.
"Pani" is somewhat like the English "ma'am/lady" or (even more so) the Spanish "señora." It's a title, and is used in a formal way to address someone directly instead of the informal "ty" ("you"). It also can be used to just mean "woman."
"Kobieta" doesn't have all those other uses as a polite pronoun or title.
Cases. There are seven cases in Polish (although we almost don't mention Vocative in this course), and you need different forms for different functions, after different prepositions, after different verbs etc.
"kobieta" is the Nominative form - the basic one, that you will find in the dictionary.
"kobietą" is Instrumental, needed for example after "to be".
"kobietę" is Accusative, which is one of the most widely used cases, for example needed after "to see".
You can take a look here and check one of the posts on Cases.