'Have' and 'have got' mean the same basic thing.
I have two brothers.
I've got two brothers.
...the same.
The question forms are:
Do you have any brothers? (Yes, I do. No, I don't.)
Have you got any brothers? (Yes, I have. No, I haven't.)
Although the meaning is the same, 'have' is a little more formal and 'have got' is a little less formal.
Have is a state verb, so it isn't usually used in the continuous form. However, here are some examples of 'having' in use:
I'm having a great time. (I'm really enjoying myself at the moment.)
I'm having a party. (I've arranged a party for some time in the future.)
I'm having a new mobile phone. (Wrong! You should say 'I have a new mobile phone.')
Brian