I'll start by correcting your question: 'What does it mean?'
And here's some answers:
1) I have a pen = correct neutral/formal
I have got a pen
I've got a pen = correct neutral/informal
I got a pen = 'incorrect'* very informal
2. I have to meet my mother. = correct
"I got to meet my mother?"*
"I gotta meet my mother" ?* or "I've gotta meet my mother" *
* You can say all of these if you want to. Millions of native speakers all over the world say these expressions the whole time. According to the grammar books, they're wrong. If you write them in an English exam, they'll be marked wrong. If you put a job application using them, you'll never get the job. If you write them in most situations, you'll give the impression that you're lazy or badly educated.
So, it's up to you. Good English or bad English. It's your choice.