I'm living with my family, and my cousin introduced me to her best friend. Her friend and I wanted to get to know each other better and found out we have everything in common. But when she told her mom about me, she said I'd be a perfect match if it wasn't for my age. I wanted to show her mom I'm not like some guys, the girl and I like each other a lot but we don't know if we can be together. She's 14 and I'm 19. What should we do?

What country is this?
Regardless, you should do what you must do by law, and that is go by the wishes of the parent(s). You are an adult and she is a minor, and any adult who even suspects that you are having sex can have you charged and ruin your life forever.
For thousands of years, it wasn't uncommon for a man your age to marry a girl about 14. Those days are gone. A 14 year old girl has school to finish with kids her own age. It isn't fair to HER to let her fall for you.
And why are you living at home? What are you doing to be an adult, either a career or further education?

No, it isn't okay, at your ages, to be involved. I don't know what a 19 year old has in common with a 14 year old girl, which is what any reasonable adult would ask- including the police.

It is normal in middle east countries and a handful of her places. You really do need to keep it as friends and nothing serious for at least a couple more years more depending where you are from

If you are 19, you should be able to date and have relationships with women your own age. Dating a 14 year old is off-limits and shows immaturity with relationships in someone your age. Are you in college or trade school? Do you work?

What should you do? Its simple, obey the wishes of her parents and have the adult sense to back off from this pursuit of their child in any form whatsoever.

Push it, and bear the consequences of your action, and for adults like yourself, that can be quite harsh, and effect YOUR whole life, and future, AND hers. Her parents have made how they feel known, so take it as a warning they will prevent any effort you make to pursue this "love".

Its pretty dumb to take a chance on the love of a 14 year old don't you think? She is jail bait to adult men.

Pursuit - that is the key, you can not pursue her as anything more than a one of many friend in your circle of friends. For one she does need time to have personal growth.

She cannot be a friend, because he has NO BUSINESS hanging out with 14 year olds unless I missed something.

I agree I would not 'hang out with'

As asked where are you. If you are in most western countries, it was irresponsible for your cousin to introduce you in that way. It was irresponsible for you to try to get to know her better.

Bottom line her mother is wrong and right. I don't believe you would be a perfect match because of your ages. So you have to go along with her parents wishes.