In the US it's most common to simply say, "Got it." This isn't standard, correct English, but it's an extremely common way to express "I understand." We usually use "I've got it" to mean that we actually "have" something. ("Where is the key?" "I've got it.") We use "got it" to say we understand. ("Don't open the email. There might be a virus." "Got it. I won't open it.") In the US, we would rarely say "I have understood," because if you "have understood" something, then you still understand it, so you can use the present tense ("I understand") to express the same meaning. You don't need to worry about the difference between present tense and present perfect tense for the phrase "got it." You never need to use present perfect for this phrase. Just "got it" is fine.

You could say "I've got it" or I got it. I don't think people would say I have got it" unless they really want to emphasise "have" when they believe the other person thinks they haven't understood.

Lily, regarding your very strange comment to Simon -- "I've" is not a preposition. There is no preposition in "I have." I agree with you that people should check the answers they receive on Italki, but you certainly don't seem to be offering helpful answers or comments. You're going to "throw your diplomas in the trash" because people are saying "got it" is okay? According to the OED, the use of "to get" with the meaning "to understand" has been in use since at least 1857. It's not some kind of crazy new slang term that just showed up in the language yesterday. The question Alex asked was about tenses. That's why the answers are focused on the tenses. And nobody is suggesting that he should say "I understand" instead of "got it" because he clearly already knows that "I understand" is the formally correct way to say it. He's asking how Americans use the phrase "got it." He wants to know how the language is actually spoken, not how to fill in the blanks on an exam.

All the questions today seem to be about slang and proper tense. Past or present. The best thing would simply be "I understand" . "I got it" is slang, but I would understand that you do understand . "I have it" is fine. "I have got it" is redundant - saying the same thing twice. "Got" and "have" both indicate possession. "I have understood" is *not* correct. Leave out "have", say "I understand" if you understand right now - in the present. Please use "understood" if you are referring to the past. Stick with proper grammar, don't worry about slang. We have enough trouble understanding each other. If you use proper grammar, you have a far better chance of being understood. If you understand what I wrote, and want to be very casual with just me? You may say "Got it, Teach!" ( "Teach" used as a nickname for teacher) and I will smile. : 0) The rest of that? A big mess. I hope that helps.