"I am sorry."
Gomen nasai = "I feel bad about it", average level of politeness Gomen, gomenne (masculine), gomenna (feminine) = "sorry dude", just for friends and often jokingly Sumimasen = "it is my responsibility", high level of politeness, can also be used as "excuse me" to formally attract attention.
I'd like to explain this a bit more, as long as it's not exactly a 100% rule. "Ne" is a particle said to search the listener agreement, making the sentences sound softer, more polite. Can be used by both genders, but it's more common between Japanese women:
"ね is a very common particle, and a polite way to end a sentence. Listen to any conversation between Japanese women and you hear lots of ね."
"Na" it's similar, but more introspective. You don't ask for the listener agreement, but for your own one. It's considered sort of rude, and definitively, it's pretty uncommon in women:
"The difference between them is that な is generally used by MALEs and may make your speech a little offensive. The polite form doesn’t generally appear in this context except for when the elderly use な."
gomenasai and sumimasen are the same when being sorry, and both are formal, believe me you can say both when you are sorry. the informal ones are suman, and gomen, gomenna gomenne, belive me there are also the highest politesness they used in japan, especially when saying sorry to customers or very high people. but for now this is good, if you want to know what those are just comment down.
This is more of a case of anime not really using normal conversational Japanese. Overly cutesy characters will often add a polite です to the end of every phrase even when it wouldn't make sense to do so.
ごめんなさい broken down is an honorific ご, 免・めん "a dismissal" and the polite imperative form of 為す・なす "do", so more literally it is a polite request/command "Do a dismissal", like saying "Forgive me". It wouldn't really make sense to add a です here.