Forms of Kufr
Just as there are different sins in Islam that differ in grade, so is Kufr. The acts of disbelief differ in damage and effect. Thus, Kufr exists in two forms: minor and the major form.
1. Minor Kufr
The minor form of Kufr, also known as Kufr Asghar or Kufr duna Kufr, is described in the Hadeeth. Those are acts that do not place a Muslim outside the fold of Islam. This sin is one of the greatest sins in Islam. It should therefore not be despised.
The concept of minor Kufr is not found in the Quran. To find it we need to dive into the Hadith, and seek the understanding of the pious predecessors.
For example, we sometimes see that an act falls under 'Kufr'. That can be both the minor or major form. By keeping to the correct context of the Hadith, we quickly find in which form an act falls.
2. Major Kufr
The deeds of major Kufr make a person leave the religion. The act is also called Kufr Akbar. These deeds differ from acts of major shirk. Major Kufr are acts of disbelief in which a partner is not nessecarily attributed to Allah.
For example, think of acts like insulting Islam or disbelieving in one of the verses from the Quran.
A person would not ascribe a partner to Allah when he contradicts the Quran for example. But its still an act of disbelief.
Thus, the distinction between major Shirk and major Kufr is that Shirk means the attribution of partners to Allah, whereas major Kufr has to do with deeds that are disbelief, without nessecarily ascribing any partners to Allah.
A doer of major Kufr is called a Kafir (disbeliever), and the doer of major Shirk is called a Mushrik (polytheist). Both are disbelievers in Islam and thus can also be called Kafir.