"I do not see this cat."
Translation:Nie widzę tego kota.
'Widzieć' requires a noun in accusative case, 'nie widzieć' - genitive. Demonstratives in Polish also declines and must agree with the noun, so they must take a correct form.
As 'kot' is animate masculine, the acc and gen are the same, and so are the demonstratives: 'Widzę/nie widzę tego kota'. A better example would be a masculine non-animate noun, for example 'stół' ('table'): 'Widzę ten stół (acc)/nie widzę tego stołu(gen)
"koty" is plural Nominative or Accusative.
"widzieć" (to see) takes Accusative. But when a verb which took Accusative is negated, it now needs Genitive. So even if the sentence was "I do not see these cats", it would be "Nie widzę tych kotów", with Genitive.
"kota" is the Genitive of "kot". It's also identical to Accusative, but that's not the place to discuss it.
The 3rd person is "widzi", not "widzie".
The 1st person is "widzę".
There is more than one pattern of conjugation. True, there are 3rd person forms that end with -ie (like "idzie"), and there are 1st person forms that end with -m ("czytam"). -ę is the most common ending though, I think.