The difference between the mushrikeen and the kuffaar, and to which category do the Jews and Christians belong?
Question

                What is the difference between mushrikeen and kuffaar? Are the Jews and Christians mushrikeen or kuffaar?.
Praise be to Allah.
    Firstly: 

    The kaafir is the one who denies and conceals the truth. The 
    basic meaning of the word kufr in Arabic is concealment. Shirk 
    means devoting worship to anyone or anything other than Allaah. 

    Kufr may take the form of denying and rejecting, but the 
    mushrik may also believe in Allaah. This is the difference between the 
    mushrik and the kaafir. 

    Each word may also carry the meaning of the other, so the 
    word kufr may be used in the sense of shirk, and the word shirk may be used 
    in the sense of kufr. 

    Al-Nawawi (may Allaah have mercy on him) said: 

    Kufr and shirk may carry the same meaning, which is disbelief 
    in Allaah, may He be exalted, or they may be used separately, whereby shirk 
    refers to the worship of idols and other created beings, whilst also 
    acknowledging Allaah, as the kuffaar of Quraysh did, and kufr may have a 
    more general meaning than shirk. End quote. 

    Sharh Saheeh Muslim. 2/71 
Shaykh ‘Abd al-‘Azeez 
    ibn Baaz (may Allaah have mercy on him) said: 

    Kufr is denial and concealment of the truth, such as one who 
    denies that prayer is obligatory, or that zakaah is obligatory, or that 
    fasting Ramadaan is obligatory, or that doing Hajj when one is able to is 
    obligatory, or that honouring one's parents is obligatory, and so on, or one 
    who denies that zina is haraam, or that drinking intoxicants is haraam, or 
    that disobeying one’s parents is haraam, and so on. 

    Shirk is devoting acts of worship to something or someone 
    other than Allaah, such as one who seeks the help of the dead, those who are 
    absent, the jinn, idols, the stars, and so on, or who offers sacrifices to 
    them, or makes vows to them. A kaafir may be called a mushrik and a mushrik 
    may be called a kaafir, as Allaah says (interpretation of the meaning): 
“And whoever invokes (or 
    worships), besides Allaah, any other ilaah (god), of whom he has no proof; 
    then his reckoning is only with his Lord. Surely, Al‑Kaafiroon (the 
    disbelievers in Allaah and in the Oneness of Allaah, polytheists, pagans, 
    idolaters) will not be successful”
    [al-Mu’minoon 23:117]

    “Verily, whosoever sets up partners (in worship) with 
    Allaah, then Allaah has forbidden Paradise to him, and the Fire will be his 
    abode”

    [al-Maa'idah 5:72]

    And Allaah says in Soorat Faatir (interpretation of the 
    meaning): 

    “Such is Allaah, your Lord; His is the kingdom. And those, 
    whom you invoke or call upon instead of Him, own not even a Qitmeer (the 
    thin membrane over the date stone).

    14. If you invoke (or call upon) them, they hear not your 
    call; and if (in case) they were to hear, they could not grant it (your 
    request) to you. And on the Day of Resurrection, they will disown your 
    worshipping them. And none can inform you (O Muhammad) like Him Who is the 
    All‑Knower (of everything)”
    [Faatir 35:13-14]

    Their calling on anything other than Allaah is called shirk 
    in this soorah, but in Soorat al-Mu’minoon it is called kufr. 

    Allaah says in Soorat al-Tawbah (interpretation of the 
    meaning): 

    “They (the disbelievers, the Jews and the Christians) want 
    to extinguish Allaah’s Light (with which Muhammad has been sent — Islamic 
    Monotheism) with their mouths, but Allaah will not allow except that His 
    Light should be perfected even though the Kaafiroon (disbelievers) hate 
    (it).

    33. It is He Who has sent His Messenger (Muhammad) with 
    guidance and the religion of truth (Islam), to make it superior over all 
    religions even though the Mushrikoon (polytheists, pagans, idolaters, 
    disbelievers in the Oneness of Allaah) hate (it)”

    [al-Tawbah 9:32-33]

    Here Allaah calls the kuffaar kuffaar, and He calls them 
    mushrikeen. This indicates that a kaafir may be called a mushrik and a 
    mushrik may be called a kaafir. There are many similar verses and 
    ahaadeeth. 

    Another example is the hadeeth in which the Prophet  
    (peace and blessings of Allaah be upon him) said: “Between a man and shirk 
    and kufr there stands his giving up prayer.” Narrated by Muslim in his 
    Saheeh from Jaabir ibn ‘Abd-Allaah (may Allaah be pleased with him). And 
    the Prophet  (peace and blessings of Allaah be upon him) said: “The 
    covenant that differentiates us from them is prayer; whoever gives it up is 
    a kaafir.” Narrated by Imam Ahmad, Abu Dawood, al-Tirmidhi, al-Nasaa’i and 
    Ibn Maajah with a saheeh isnaad from Buraydah ibn al-Husayb (may Allaah be 
    pleased with him). And Allaah is the Source of strength. End quote. 

    Majmoo’ Fataawa al-Shaykh Ibn Baaz, 
    9/174, 175. 

    The Shaykh (may Allaah have mercy on him) also said:  

    It is also shirk to worship only something other than Allaah. 
    This is called shirk, and it is called kufr. Whoever turns away from Allaah 
    altogether and devotes his worship to something other than Allaah, such as 
    trees, rocks, idols, the jinn or some of the dead such as those whom they 
    call awliya’ (“saints”), and worship them, pray to them or fast for them, 
    and forget Allaah altogether, this is the worst form of kufr and shirk. We 
    ask Allaah to keep us safe and sound. Similarly those who deny the existence 
    of Allaah and say that there is no god, that life is only material, such as 
    the communists and atheists, are the most disbelieving and misguided of 
    people, and the worst in terms of shirk. We ask Allaah to keep us safe and 
    sound. The point is that people who hold these and similar beliefs are all 
    regarded as mushrikeen and kaafirs who disbelieve in Allaah. Some people, 
    out of ignorance, mistakenly regard calling upon the dead and seeking their 
    help as waseelah and think that it is permissible. This is a serious error, 
    because this action is one of the worst forms of shirk and associating 
    others with Allaah. Some ignorant people and mushrikeen call it waseelah, 
    but it is the religion of the mushrikeen whom Allaah criticized and 
    condemned. He sent the Messenger and revealed His Books to denounce it and 
    warn against it. End quote. 

    Majmoo’ Fataawa al-Shaykh Ibn Baaz, 
    4/32, 33 

    Secondly: 
The Jews and Christians 
    are both kaafirs and mushrikeen. They are kaafirs because they deny the 
    truth and reject it. And they are mushrikeen because they worship someone 
    other than Allaah. 

    Allaah says (interpretation of the meaning): 
“And the Jews say: ‘Uzair 
    (Ezra) is the son of Allaah, and the Christians say: Messiah is the son of 
    Allaah. That is their saying with their mouths, resembling the saying of 
    those who disbelieved aforetime. Allaah’s Curse be on them, how they are 
    deluded away from the truth!

    31. They (Jews and Christians) took their rabbis and their 
    monks to be their lords besides Allaah (by obeying them in things which they 
    made lawful or unlawful according to their own desires without being ordered 
    by Allaah), and (they also took as their Lord) Messiah, son of Maryam 
    (Mary), while they (Jews and Christians) were commanded [in the Tawraat 
    (Torah) and the Injeel (Gospel)] to worship none but One Ilaah (God — 
    Allaah) Laa ilaaha illa Huwa (none has the right to be worshipped but He). 
    Praise and glory be to Him (far above is He) from having the partners they 
    associate (with Him)”

    [al-Tawbah 9:30, 31]

    Here they are described as mushrikeen. In Soorat al-Bayyinah 
    they are described as kaafirs, as Allaah says (interpretation of the 
    meaning): 

    “Those who disbelieve from among the people of the 
    Scripture (Jews and Christians) and Al‑Mushrikoon, were not going to leave 
    (their disbelief) until there came to them clear evidence”

    [al-Bayyinah 98:1]

    Shaykh ‘Abd al-‘Azeez ibn Baaz (may Allaah have mercy on him) 
    said, refuting those who say that the word mushrikeen cannot be applied to 
    the People of the Book: 

    It is most likely that the people of the Book are included 
    among the mushrikeen, men and women alike, when this word is used in general 
    terms. because the kuffaar are undoubtedly mushrikeen. Hence they are 
    forbidden to enter al-Masjid al-Haraam, because Allaah says (interpretation 
    of the meaning): 

    “O you who believe (in Allaah’s Oneness and in His 
    Messenger Muhammad)! Verily, the Mushrikoon (polytheists, pagans, idolaters, 
    disbelievers in the Oneness of Allaah, and in the Message of Muhammad) are 
    Najasun (impure). So let them not come near Al‑Masjid Al-Haraam (at Makkah) 
    after this year”

    [al-Tawbah 9:28]
If the People of the 
    Book did not come under the general heading of mushrikeen, then this verse 
    would not apply to them, and Allaah would not have referred to the beliefs 
    of the Jews and Christians in Soorat Baraa’ah (al-Tawbah) where He says 
    (interpretation of the meaning): 

    “they (Jews and Christians) were commanded [in the Tawraat 
    (Torah) and the Injeel (Gospel)] to worship none but One Ilaah (God — 
    Allaah) Laa ilaaha illa Huwa (none has the right to be worshipped but He). 
    Praise and glory be to Him (far above is He) from having the partners they 
    associate (with Him)”

    [al-Tawbah 9:31]
So they are all 
    described as mushrikeen, because the Jews said that ‘Uzayr is the son of God 
    and the Christians said that the Messiah is the son of God; and because they 
    took their priests and rabbis as lords instead of Allaah. All of this is the 
    worst form of shirk. And there are many similar verses. End quote. 

    Majmoo’ Fataawa al-Shaykh Ibn Baaz, 
    4/274 

    And Allaah knows best.