Have/have got

Form
I have got a cat / I have a cat.
I haven't got a cat / I don't have a cat.
Have you got a cat? / Do you have a cat?

"Have you a cat?" and "I haven't a cat" are possible but not common

Have got is more common and sounds more natural in informal language. It is a present tense of 'have', not the present perfect of get.
Functions and examples

We use 'have got' and 'have' to talk about possession.
I've got a new house / I have a new house.
Has she got a car? / Does she have a car?

We use 'have got' and 'have' to talk about relationships.
Have you got a girlfriend? / Do you have a girlfriend?
He's got three brothers / He has three brothers.

We use 'have got' and 'have' to talk about illnesses.
I've got a bad cold / I have a bad cold.

We use 'have got' and 'have' to talk about characteristics.
Her office has got a nice view / Her office has a nice view.
Why has he got a tattoo? / Why does he have a tattoo?
Important points

With the functions above, 'have got' and 'have' cannot be used in continuous form.
I have got / have a headache.
I'm having a headache. x

In the past, 'have' is more common than 'have got'
She had a pink guitar when she was young.
She had got a pink guitar when she was young. x

Did you have a headache yesterday?
Had you got a headache yesterday? x