Generally the word kafir means 'unbeliever' and it is not meant as a derogatory label (unless it is used against Muslims by another Muslim). Of course, there are different levels of Kufur (disbelief) so although a person may be correctly identified as a kafir, that person is not kafir in the strongest sense of the word. Absolute kufur is the denial of God. Here are some examples of kufur:
1) A person is known as a Kafir if they are disbelievers in Allah, in His Oneness, and in His final Messenger, Muhammad (SAW). (E.g. An apostate from Islam)
(2) A 'Muslim' who disbelieves in a necessary tenant of Islam like the 5 pillars or some other things is known as a disbeliever.
(3) An originally non-Muslim person who denies the religion of Islam while knowing in his heart that it is true.
(4) A disbeliever in Islam only because they originally follow another (monotheistic) faith.
(5) A person who follows another (monotheistic) faith and knows nothing about Islam.
(6) A person who believes in many gods (polytheist).
(7) An agnostic who is a monotheist is a lesser kafir than a polytheist.
(8) An Atheist - ultimate kufur.
Generally a Christian or Jew is not identified as a kafir but they are known respectfully as Ahlul kitab - People of the Book(s), and Islam considers their religions valid but once they find Islam they are expected to follow the updated covenant sent by God.
وَإِذْ قُلْنَا لِلْمَلاَئِكَةِ اسْجُدُواْ لآدَمَ فَسَجَدُواْ إِلاَّ إِبْلِيسَ أَبَى وَاسْتَكْبَرَ وَكَانَ مِنَ الْكَافِرِينَ