For masculine singular it is taken into consideration if the noun is animate or not - animate has acc=gen (Widzę konia, psa, strażaka) , non-animate has acc=nom (Widzę stół).
For masculine plural - if the noun is personal or not - personal has acc=gen (Widzę strażaków), non-personal has acc=nom (Widzę psy, stoły, konie).
So the horses jump into the second group when in plural ;)
It's mostly an orthographical thing. The nominative for horse is „koń”. If it were „kon”, then indeed the genitive form would be „kona”. In case of „koń” you would want to also add „-a” and you would get „końa”… which is nice and all, but Polish orthography rules don't allow you to use consonants with a line above them right in front of a vowel (if it's in the same word). In cases like this you have to use „ni” instead (or just „n”, if it is in front of „i”).