Arabic Root:
From the root ayn-zay-zay (ع ز ز), which has the following classical Arabic connotations: to be mighty, potent, strong, powerful to be high, elevated, noble, exalted to be honorable, noble, glorious, illustrious to be indomitable, invincible to be respected, cherished, dear to be scarce, rare, precious, and unattainable.

There are several meanings to the name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ as Al-Aziz (Al-Azeez). Common translations include The Almighty, The Mighty One, or The Eminent One. The word Azeez comes from izzat meaning honor and nobility.

The divine pairing of names:
This great name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ appears approximately 90 times in the Qur'an. It's often paired with another name, for example, Al-Aziz Al-Rahim, Al-Aziz Al-Hakeem, Al-Aziz Al-Ghaffar, or Al-Aziz Al-Wahhab. The pairing of names oftens conveys an additional layer, we learn something new about Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ that would often be overlooked if we just studied these names independently.

For example, what does it mean to be Al-Aziz Al-Rahim? Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is in a position of power, He can subdue anyone who opposes Him. Yet to those who deny and to those who disbelieve, He is always merciful first. “My mercy predominates My wrath” He does not hasten to punish the one who errs but gives them time to repent. And even when He punishes the transgressors, He can be most severe, but He punishes with mercy. These meanings are said to be divinely designed, whose perfection in meaning is understood best through the pairing of names.

أَمۡ عِندَهُمۡ خَزَآئِنُ رَحۡمَةِ رَبِّكَ ٱلۡعَزِيزِ ٱلۡوَهَّابِ

Am’indahum khazaaa ‘inu rahmati Rabbikal Azeezil Wahhab

English Translation:
"Or do they have the depositories of the mercy of your Lord, the Exalted in Might, the Bestower?"— (Qur'an 38:9)

The Pairing Al-Azeez and Al-Hakeem:

وَلِلَّهِ جُنُودُ ٱلسَّمَٰوَٰتِ وَٱلۡأَرۡضِۚ وَكَانَ ٱللَّهُ عَزِيزًا حَكِيمًا

Wa lillaahi junoodus samaawaati wal ard; wa kaanal laahu azeezan hakeema

English Translation:
"And to Allah belong the soldiers of the heavens and the earth. And ever is Allah Exalted in Might and Wise."— (Qur'an 48:7)

The juxtaposition of Al-Aziz and Al-Hakeem is also an interesting pairing. The saying "absolute power corrupts absolutely" does not apply to Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ as He is the all-wise. Humans are fallible. We see it repeatedly happen when a person gains absolute power in their small domain. It's often met with cronyism, nepotism, and corruption. A person you perceive as wise or humble can become blinded by power. They are not all-wise and all-knowing. Only Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is the rare exception capable of combining all these attributes perfectly.

فَالِقُ ٱلۡإِصۡبَاحِ وَجَعَلَ ٱلَّيۡلَ سَكَنٗا وَٱلشَّمۡسَ وَٱلۡقَمَرَ حُسۡبَانٗاۚ ذَٰلِكَ تَقۡدِيرُ ٱلۡعَزِيزِ ٱلۡعَلِيمِ

Faaliqul isbaahi wa ja’alal laila sakananw wash shamsa walqamara husbaanaa; zaalika taqdeerul Azeezil Aleem

English Translation:
"[He is] the cleaver of daybreak and has made the night for rest and the sun and moon for calculation. That is the determination of the Exalted in Might, the Knowing."— (Qur'an 6:96)

وَمِنَ ٱلنَّاسِ وَٱلدَّوَآبِّ وَٱلۡأَنۡعَٰمِ مُخۡتَلِفٌ أَلۡوَٰنُهُۥ كَذَٰلِكَۗ إِنَّمَا يَخۡشَى ٱللَّهَ مِنۡ عِبَادِهِ ٱلۡعُلَمَـٰٓؤُاْۗ إِنَّ ٱللَّهَ عَزِيزٌ غَفُورٌ

Wa minan naasi wadda waaabbi wal an’aami mukhtalifun alwaanuhoo kazalik; innamaa yakhshal laaha min ‘ibaadihil ‘ulamaaa’; innal laaha Azeezun Ghafoor

English Translation:
"And among people and moving creatures and grazing livestock are various colors similarly. Only those fear Allah, from among His servants, who have knowledge. Indeed, Allah is Exalted in Might and Forgiving."— (Qur'an 35:28)

*There are more pairings but the most common ones are listed above.

Reflection:
How can we benefit from knowing that Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Aziz? We can supplicate using this name if we wish to be mighty. It's natural to aspire for wealth, success, or prestige. It's a noble cause, especially if the goal of attainment is using that wealth to further Islam and/or to help others. To be blinded in the pursuit of these things that you forget all your other duties is not virtuous and should be avoided. It's a fine line that requires careful examination, honesty with oneself, and awareness.

If you have hunger or burning desire for success, health, wealth, and prestige you should seek it from the true possessor of might, honor and rank. In Surah Imran ayah 26:

قُلِ ٱللَّهُمَّ مَٰلِكَ ٱلۡمُلۡكِ تُؤۡتِي ٱلۡمُلۡكَ مَن تَشَآءُ وَتَنزِعُ ٱلۡمُلۡكَ مِمَّن تَشَآءُ وَتُعِزُّ مَن تَشَآءُ وَتُذِلُّ مَن تَشَآءُۖ بِيَدِكَ ٱلۡخَيۡرُۖ إِنَّكَ عَلَىٰ كُلِّ شَيۡءٖ قَدِيرٞ

Qulil laahumma Maalikal Mulki tu’til mulka man tashaaa’u wa tanzi’ul mulka mimman tashhaaa’u wa tu’izzu man tashaaa’u wa tuzillu man tashaaa’u biyadikal khairu innaka ‘alaa kulli shai’in Qadeer

English Translation:
"Say, “O Allah, Owner of Sovereignty, You give sovereignty to whom You will and You take sovereignty away from whom You will. You honor whom You will and You humble whom You will. In Your hand is [all] good. Indeed, You are over all things competent."— (Qur'an 3:26)

References:
01) Sahih Muslim 2751a