When can "have" be used without "got"?

I read this article and now I'm confused when got can be omitted when using have.

Could this be explained in plain English without technical terms?

Is there a different usage in past tense?

You can safely omit it pretty much always.

I think it's largely a regional variety issue. As the article explains, in the UK "I've got" might be used more commonly where Americans would say "I have".

I remember being taught phrases like this in elementary English classes (due to British English emphasis in Finnish schools at that time):

I've got a cat.

But rest assured, it is never wrong to omit "got" in a phrase like this and just say:

I have a cat.

Also in those cases where it's used to add emphasis or indicate obligation...

I've got to go now!

... you can always omit it and still be perfectly understood (you can add the emphasis in other ways):

I have to go now!