There are a number of types of Kufr mentioned in the Quran.
The root letters are ك ف ر which mean to cover or hide. Kufr means blasphemy or disbelief and Kaafir means disbeliever.
Kufr al Akbar
Those who have received information of the truth yet out rightly deny the message and Allah himself and His Dominion and Authority. This is the opposite to imaan, or belief.
Within this category there are others
Kufr ul-‘Inad
This is disbelief out of stubbornness. This applies to someone who knows the truth and admits to knowing the truth and admits to knowing it with his tongue, but refuses to accept it and refrains from making a declaration.
Allah ﷻ says, “Throw into Hell every stubborn disbeliever.”
Surah Qaf (50), ayah 24
Kufr ul-Inkar
This is disbelief out of denial. This applies to someone who denies with both heart and tongue.
Allah ﷻ says, “They recognise the favours of Allah, yet they deny them. Most of them are disbelievers.”
Surah Nahl (16), ayah 83
Kufr ul-Kibr
This is disbelief out of arrogance and pride. The disbelief by the devils (e.g. Iblis) is an example of this type of kufr.
Kufr ul-Juhud
This is disbelief out of rejection. This applies to someone who acknowledges the truth in his heart, but rejects it with his tongue. This types of kufr is applicable to those who calls themselves Muslims but who reject any necessary and accepted norms of Islam such as salat and zakat.
Allah ﷻ says, “They denied them (i.e. the signs) even though their hearts believed in them, out of spite and arrogance.”
Surah Naml (27), ayah 14
Others include:
Kufr ul-Nifaq
This is disbelief out of hypocrisy. This applies to someone who pretends to be a believer but conceals his disbelief. Such a person is called a munafiq or hypocrite.
Allah ﷻ says, “Verily the hypocrites will be in the lowest depths of Hell. You will find no one to help them.”
SurahAn-Nisa (4), ayah 145
Kufr ul-Istihal
This is disbelief out of trying to make haram into halal. This applies to someone who accepts as lawful (halal) that which Allah has made unlawful (haram) like alcohol or adultery. Only Allah ﷻ has the prerogative to make things halal and haram and those who seek to interfere with His right are like rivals to Him and therefore fall outside the boundaries of faith.
Kufr ul-Kurh
This is disbelief out of detesting any of Allah’s Commands.
Allah ﷻ says, “Perdition (destruction) has been consigned to those who disbelieve and He will render their actions void. This is because they are averse to that which Allah has revealed so He has made their actions fruitless.”
Surah Muhammad (47), ayah 8-9
Kufr ul-Istihzaha
This is disbelief due to mockery and derision.
Allah ﷻ “Say: Was it at Allah, His signs and His apostles that you were mocking? Make no excuses. You have disbelieved after you have believed.”
Surah Taubah (9), ayah 65-66
Kufr ul-I’radh
This is disbelief due to avoidance. This applies to those who turn away and avoid the truth.
Allah ﷻ says, “And who is more unjust than he who is reminded of his Lord’s Signs but then turns away from them. Then he forgets what he has sent forward (for the Day of Judgment).”
Surah Kahf (18), ayah 57
Kufr ul-Istibdal
This is disbelief because of trying to substitute Allah’s Laws. This could take the form of (a) rejection of Allah’s Law (shari’ah) without denying it, (b) denial of Allah’s law and therefore rejecting it, or (c) substituting Allah’s laws with man-made laws.
Allah ﷻ says, “Or have they partners with Allah who have instituted for them a religion which Allah has not allowed.”
Surah Shura (42), ayah 8
Allah ﷻ says, “Say not concerning that which your tongues put forth falsely (that) is lawful and this is forbidden so as to invent a lie against Allah. Verily, those who invent a lie against Allah will never prosper.”
Surah Nahl (16), ayah 116
This was taken from Tafsir ibn Kathir