actually this word is gramatically masculine, but can describe a horse of unspecified gender, if you want to be specific - mare is "klacz" , stallion is "ogier", gelding horse is "wałach"

Cases. "koń" is the basic, Nominative form, "konia" is Accusative. The verb "lubić" (to like) takes Accusative.

we do not have word for "the", so sometimes we use forms of "ten/tamten" , but usually we skip it.

I also don't remember having a lesson on the tamtego group of words, but at one stage they started to appear and I gradually understood that they mean "that", and differ by gender, number etc. My brain too, felt it couldn't cope, as I couldn't work out the logic behind which form was which, and I just had to keep guessing. Following that, the group of similar words without the "tam" prefix also appeared, and I soon learnt (by the effective "stick" method of having to keep repeating exercises!) that they mean "this". So they both sort of stand in for "the" but are more specific. An example of how we could do with more "tips and notes" on grammar?

They're in Accusative, which for masculine animate nouns looks just like its Genitive. "lubić" takes Accusative.

If you negated it, the sentence would only differ by "Nie" ("Nie lubię tamtego konia"), but in fact Accusative would have changed into Genitive here.