What you're looking for is chat with end-to-end encryption — where only you and the recipient can decrypt the messages.
To use the passing-a-note-in-class analogy, you and your friend both have a unique way to confirm you are who you say you are — in real life, handwriting; on the internet, a public key.
The note is then encrypted using a randomly-generated cipher, which then is also encrypted and can only be decrypted with the receiver's private key.
Admittedly, the analogy kinda breaks down here, but the end result is that this system ensures that messages sent to a single person can only be read by that person only.
Which is best?
If unflinching privacy is your aim, then Open Whisper Systems's Signal, is the industry gold-standard.
Oh and did I mention that it is completely free?
You really should check it out.