Azeez can be derived from one of the following words:

A. Azeez refers to something matchless and unique. When something is scarce, the Arabs say :

عز الطعام في البلد
“Food is scarce in the city.”

So, when the creation is referred to as Azeez, when they are scarce, then Allaah (Jalla Jalaalahoo) is Most Worthy of being referred to as Azeez, as there is nothing similar to Him.  

B.  Azeez means dominant and overpowering. Arabs say :

من عز بر اي من غلب سلب
“Whoever dominates, will gain the booty.”

When a weak human overpowers his opponent, he is called Azeez. Allaah (Jalla Jalaalahoo) can never be overpowered.
He is Most Worthy of being referred to as Azeez. Imaam Ibn Jareer Rahimahullaah says :

اَلشَّدِیْدُ فِي انْتِقَامهمِ مَّنِ انْتَقَمَ مِنْ أَعْدَائهمِ مَّنْ أَرَادَ الإِنْتِقَامَ مِنهُ لَایَقْدِرُ أَحَدٌ یَدْفَعه عَنه
“ Al-Azeez is that Being Who is Severe in Punishing His enemies. When He Intends to Punish someone, there is no power which can ward it off.”

C. Azeez refers to a Being with Great Power. Al-Azeez is that Being Who Possesses every type of Greatness. He Possesses Strength as well as Domination. All occurrences are within His Choice. 
It is not possible for any of the creation to achieve any of His Qualities. He Dominates over all things in existence. In the face of His Grandeur, all creation is completely helpless and powerless.

Imaam Qurtubi Rahimahullaah says,
“Al-Azeez is that Being whose Power, Strength, Domination and Subjugation, encompasses every single thing, in such a way, that nothing can make Him helpless, nor can anything defeat Him. Nothing lies out of the framework of His Strength and Power. None can escape from His Grip, nor can anyone overpower Him.”

D.  Azeez refers to the Being Who Grants Honour. All Honour Belongs Solely to Him. 

Allaamah Teebee Rahimahullaah has stated,
“Al-Azeez is that Being, Who, after diving into the oceans of His Greatness , intellects are dumbfounded; brains are astounded in attempting to comprehend His Attributes, tongues become exhausted in attempting to extol His Splendour and Grandeur.”

Imam Ghazaali Rahimahullaah has stated,
“Azeez is a Great Being whose like is none, all are in need of him, and reaching him is difficult. Whoever does not possess all of these three qualities cannot be referred to as Al-Azeez.”

There are many things whose existence is minimal, but are not great, and very little benefit is derived therefrom. Then there are many whose status is great, and whose benefit is abundant, and no equal is found, but if it is not difficult to reach them, they cannot be called Azeez. An example of this is the sun and earth.

They have no equal, and there is great benefit in both, and all are in need of them. However, they cannot be called Azeez, as it is not difficult to see them.  

In each of these three qualities, there is excellence and deficiency. Perfection in minimal is when there is no one else, and something similar is impossible to be found. This is only for Allaah (Jalla Jalaalahoo). Even though the sun is one, there is a possibility of another one, or an even better one being created. 

The extreme level of need is when everything is in need of one, in every aspect, even in one’s existence, qualities and permanence. This too, is solely found in the Being of Allaah (Jalla Jalaalahoo). The perfection of the difficulty in reaching Him, is that reaching Him is impossible, which means that His Reality can never be comprehended. 
None can recognize Allaah (Jalla Jalaalahoo), but Allaah (Jalla Jalaalahoo). So, He is the Only Being, who is Al-Azeez in reality.