How shoud I say: "I've got it" or "I got it"?

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I've got it (Mainly British)

I got it (Mainly American)

-so they are both right-

Americans often omit present perfect when referring to a recent past.

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It depends on the context. 
'I got it' could mean I bought it or I took it. It can also mean the you understand something. 
'I've got it' means I have it.

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I've got it - подходит скорее для ситуации, когда ты уже знал ту информацию, которую тебе рассказал собеседник, а "I got it" это аналог русского выражения "я понял"

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You got that? :P

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Well, if you say "I've got it" the 've is not pronounced very much, like the n in "Bonjour" isn't. 
So "I've got it" is second best to "I got it!" (USE "I GOT IT!") :)

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