Facebook Messenger shows the conversations you had with your friends or other people. If you want to remove someone from Facebook, unfriend him. If you want to remove the conversation, swipe left across the conversation you want to delete and then tape Delete.
To delete someone from facebook messenger you want to click on the I button then click on the three dotted button on the top right then click on delete conversation.
If you want them to completely delete them off facebook messenger, you have to defriend them