"I do not like that woman."
Translation:Nie lubię tamtej kobiety.
"lubić" (to like) takes Accusative. So first you have Nominative "tamta kobieta" ('that woman' as the subject of the sentence, for example), then you have Accusative "tamtą kobietę", and finally Genitive "tamtej kobiety" - as every time a verb needing Accusative is negated, it then needs Genitive.
As for 'tej' and 'tamtej', it's the thing with the different perception of closeness. Polish has "ta/ta/tamta" (and forms), while English has "this/that/that". So the second 'ta' overlaps with the first 'that'. That's why "that woman" can be understood as "ta kobieta" as well.
In this explanation "So while the direct translation of "tego" is of course "this", "that" is perfectly acceptable. In fact, sometimes native English speakers complain that the English sentence is unnatural, because it uses "this" and not "that"... but in Polish in such a sentence "tego" is perfectly natural. Like here. Some people could argue that 'not seeing "this" horse' doesn't make much sense, that it should be "that horse"." you explained the interchangeability of this/ that so why wouldn't tej/ tamtej be equally accepted in this question.
Yeah, they are, but they are different cases. "tamta" is Nominative. Like "tamta kobieta" - used for the subject of the sentence.
Then you have "tamtą kobietę" (Accusative), generally used for the direct object. Needed by numerous verbs and prepositions.
This here ("tamtej kobiety") is Genitive. Also needed by various verbs and prepositions, just not that numerous... And why is it used here? Well, "lubić" (to like) takes Accusative. But when a verb needing Accusative gets negated, it takes Genitive instead. This is the only case that changes when negated, other cases just stay the same.
I do not like translates to Nie lubię. The conjugated Polish verb already contains information about the subject, so the subject pronoun ja is not required.
Having said that, we do also accept Ja nie lubię.... You probably made a mistake somewhere else and got redirected to the main answer, which omits the pronoun.