How should we
    respond when the People of the Book greet us with salaam?
Question

                What is the proper way of greeting a
      non-muslim (5%'er,FOI,or Christian)when they greet you with the correct Salam's.
Praise be to Allah.If the Muslim realizes that the kaafir has said to him al-saam 
      alaykum, which means may death be upon you, he should respond 
      in kind by saying wa alaykum (and also upon you).
If he realizes beyond a doubt that he has greeted him with 
      the greeting of Islam (al-salaamu alaykum), Ibn al-Qayyim (may Allaah have 
      mercy on him) said: if the listener realized that the Dhimmi [Jew or Christian] 
      has said salaam alaykum and he is sure of that, should 
      he say wa alayk al-salaam or just wa alayk?
According to the evidence and principles of shareeah, 
    he should say wa alayk al-salaam, because this is more fair, 
    and Allaah commands us to be just and to treat others well.
(Ahkaam Ahl al-Dhimmah, 1/425, 426).
Shaykh Muhammad al-Saalih ibn Uthaymeen (may Allaah 
      preserve him) said:
 These people who have come to us from the 
    east and the west and are not Muslims, it is not permissible for us to initiate 
    the greeting of salaam with them, because the Prophet  
    (peace and blessings of Allaah be upon him) said: Do not initiate the 
    greeting of salaam with the Jews and Christians. (Narrated by 
    Muslim in his Saheeh).
If they greet us, then we can respond in a manner similar 
    to that in which they greet us, because Allaah says (interpretation of the 
    meaning):
When you are greeted with a greeting, greet in return 
    with what is better than it, or (at least) return it equally [al-Nisaa 
    4:86].
When they greet us with the greeting of Islam  al-salaamu 
    alaykum  either of the following two cases applies:
Either they pronounce the laam (l-sound) 
    clearly, and say al-salaam alaykum (peace be upon you), 
    so we may say wa alaykum al-salaam or wa alaykum
Or they do not pronounce the laam clearly, 
    and so they say al-saamu alaykum (death be upon you), so 
    we should say wa alaykum only. This is because the Jews 
    used to come to the Prophet  (peace and blessings 
    of Allaah be upon him) and greet him by saying al-saam alaykum, 
    without pronouncing the laam. Al-saam means death; 
    i.e., they were praying against the Prophet  (peace 
    and blessings of Allaah be upon him), praying that he would die. So the Prophet 
     (peace and blessings of Allaah be upon him) commanded 
    us to say to them wa alaykum.
Thus, if they say al-saam alaykum, we should 
    reply, wa alaykum, which means: and the same to you, may 
    death be upon you. This is what is indicated by the Sunnah.
But if we initiate the greeting of salaam with them, our 
    Prophet  (peace and blessings of Allaah be upon 
    him) forbade us to do this.
(Majmoo Fataawaa Ibn Uthaymeen, 2/97, 
    98).
And Allaah knows best.
For more information, see Question # 6583.