Calm down
Means "calm" or "be calm". Use 'tranquilo' for a man and 'tranquila' for a woman. It's a very common, slightly softer way to tell someone to calm down.
Means "relax". Use this just like in English when someone is tense or stressed.
Means "Don't worry". Use this to reassure someone who is anxious about a specific problem.
Means "Everything is okay". This is a comforting phrase to help someone see that the situation is not as bad as they think.
Calm down, please. We are going to solve this together.
Calm down (to a female), the bus arrives in five minutes. We aren't going to be late.
Relax! It's just a game. It's not that important.
Don't worry about the test. You studied a lot.
'Cálmate', 'tranquilo/a', and 'relájate' are universally understood and used.
They use the same phrases. When speaking to a group of friends, they use the 'vosotros' form: 'calmaos' and 'relajaos'.
Due to the use of 'vos', you will hear 'calmate' (stress on the second 'a') and 'relajate'. Also, the word 'tranqui' is a very common informal shortening of 'tranquilo'.
A very colloquial, informal way to say this is 'bájale dos rayitas', which literally means 'lower it two little lines', similar to 'take it down a notch'.
It's common to hear 'fresco' used like 'tranquilo'. For example, 'Fresco, todo está bien' means 'Chill, everything is okay'.
Saying 'Calma' instead of 'Cálmate'. — 'Calma' is a noun ('calmness'). The command form requires the reflexive pronoun '-te' attached to the verb: 'Cálmate'. You are telling the person to calm *themselves*.
Forgetting gender with 'tranquilo'. — This is an adjective, so it must agree with the person you are talking to. Say 'tranquilo' to a man and 'tranquila' to a woman.
Using the formal 'cálmese' with a friend. — 'Cálmate' is for 'tú' (informal you). 'Cálmese' is for 'usted' (formal you). Using the formal version with a friend can sound distant or even sarcastic. Stick to 'cálmate' with peers.