How can I make my Tweets private so that they can only be viewed by the person I am sending them to?
You need to use Direct Messages.
You need to follow the other person and they need to follow you, then you can send each other DMs and no one else will see them.
Unless you protect your Tweets all Tweets are public by default.
Even when you protect your Tweets all your followers will be able to see any Tweet you post.