1. 'Get' is a verb that has three main meanings - to obtain, to become and to arrive. It can also combine with other words to create a variety of meanings - get up, get out, get round and so on.
The present simple form of this verb is 'get' (or 'gets' in the third person), and the past simple is 'got'.
Present: I get home at 6 o'clock every day (arrive)
Past: I got home at 6 o'clock yesterday (arrived)
2. The word 'got' can also be an extra little word which is added to the verb 'have' in the present tense.
'I've got a problem' = 'I have a problem'
There's no difference in meaning between 'I've got a problem' and 'I have a problem'. We just contract the 'have' to 've' and add 'got' because it's easier to say, and more natural and informal than 'I have'. Here's my advice - don't try to analyse the grammar of 'got' when it's used in this way, and don't worry about the fact that it looks like a past tense. Just accept it as an idiomatic feature of informal English.