'I've got' has the same meaning 'I have', e.g. 'I've got 2 dogs' = 'I have 2 dogs'

Also, 'I've got to...' is the same as 'I have to...' e.g. 'I've got to work tomorrow' = 'I have to work tomorrow'

'I got' is the simple past of 'to get' e.g. 'I got a new bike for Christmas.' It is also used in the passive past tense, as the other answerer said, e.g. 'last week I got arrested', 'yesterday he got mugged' (I think in all instances of this, 'was/were' can be used instead of 'got' - 'got' is slightly informal)

However 'I got' is also very commonly used in US English as just a shorter way of saying 'I've got' - I hear it all the time in American music/TV etc! e.g. 'I got a problem' = 'I've got/I have a problem'

Thanks a lot Alemão com Johannes! this is very helpful.

"I've got" signals possession. For example, I've got a headache

"I got" is the past tense passive, like I got arrested

Thank you Emily! This helped me a lot too! :)