These are not idiomatic expressions, Rigoberto, they are just two forms of the informal stative verb 'have got' = have. The first, 'I got' is substandard, and the second, 'I've got' is informal, for 'have' (= possess), as in these sentences:
'I have a headache.'
'I've got (have got) a headache.'
X 'I got a headache.'
The forms have other uses, however, as the past tense and present perfect tense for the active verb, 'to get':
'I got a letter from my friend yesterday.'
'I've got (BrE)/gotten (AmE) a letter from her every day this week.'
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