There are many different views in Islam, but all Muslims have major things in common:
Belief in Allah as the one and only God who created the world
Belief in Quran as the messages of Allah to the human
Belief in prophet Muhammad (PBUH) as the last prophet sent by Allah
Belief in Ka'ba as the unique Qibla to pray towards
Belief in all the previous prophets including Adam, Noah, Ibrahim, Musa and Isa
Belief in the future life, the day of judgement, heaven and hell
And many other things
So this is my question: If there is another group of people who believe in Allah as the unique God, Prophet Muhammad (PBUH) as the last prophet, and the holy Quran, is another Muslim allowed to call them unbeliever (Kafir)? - By Kafir I mean Kufr-Ul-Kibr as stated in @DanAndrew answer below.
Common people may not issue the verdict of Kufr on a Muslim, unless he explicitly confirms it.
A Muslim may be described as "doing an action of Kufr" if that is clearly the case, but that does not make him a Kafir. An example of such an action would be to abandon praying completely.
To claim Kufr of a Muslim, an Islamic judge has to go through a number of very strict steps to confirm it. For example, in the aforementioned case of abandoning prayers, the judge has to make sure the accused understands all the following:
The accusation and its cause.
The rules that make performing this action obligatory.
The proofs (Quran, Hadith, Tafseer, Fiqh, etc.) that make abandoning this action an action of Kufr.
The accused has to confess that he denies the obligation of such an action, and that he is not leaving it out of laziness for example.
The rules above are neither comprehensive nor decisive, but you get the idea.
Types of kufr (disbelief) - Adapted from 'Tafseer ibn Katheer The
Qur'an uses the word kufr to denote a person who covers up or hides
realities, one who refuses to accept the dominion and authority of God
(Allāh). There are several types of Al-Kufr al-Akbar:
Kufrul-Kibr: Disbelief out of arrogance and pride. The disbelief by the devils (Iblees) is an example of this type of Kufr.
Kufrul-Juhood: 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
Salaat and Zakat. Allaha says: They denied them (OUR SIGNS) even
though their hearts believed in them, out of spite and arrogance.
[Soorah Naml (27), Ayah 14]
Considering someone Kufrul-Kibr should be used with great caution as it is extremely offensive. For example, I have a Muslim friend who drinks beer. He knows he shouldn't, but he does (and feels bad). This is not a case for Kufrul-Kibr.
Like any library, Islam Stack Exchange offers great information, but does not offer personalized advice or legal verdicts, and does not take the place of seeking such advice from your Mufti, Imam, or other trustworthy Counselor.