(The Mighty One)
The Strong, The Defeater, The One Who is not Defeated.
Al-Aziz (in Arabic: ٱلْعَزِيزُ) is the one who is most powerful and all-strong. He is the conqueror that is never conquered and never harmed. This is because His strength, glory, and power are overwhelming, and cannot be overcome or resisted. He is the only one with complete mastery over all creation. The One who is the cherished source of all strength, power, and potency.
From the root '-z-z 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.
Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is al-Aziz (al-Azeez), common translations include the almighty, the mighty one, or the eminent one. He is omnipotent over all his creation.
This great name of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ appears approximately 92 times in the Qur'an. It's often paired with another name, for example, al-Aziz al-Rahim, al-Aziz al-Ghaffar, or al-Aziz al-Wahhab. This pairing is quite impactful, 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 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 He gives them time to repent. As reference below you can view the other mentions of al-Aziz from the Qur'an.
Am’indahum khazaaa ‘inu rahmati Rabbikal ‘Azeezil Wahhab
"Or do they have the depositories of the mercy of your Lord, the Exalted in Might, the Bestower?" (Qur'an 38:9)
Wa lillaahi junoodus samaawaati wal ard; wa kaanal laahu ‘azeezan hakeema
"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 happen repeatedly when a person gains absolute power, it's coupled with cronyism, nepotism, and corruption. A person who you perceive as wise or humble easily gets blinded by absolute power. They are not all-wise and all-knowing. Only Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is the rare exception capable of combining all these attributes which is why it's befitting we call Him the all-Mighty.
Faaliqul isbaahi wa ja’alal laila sakananw wash shamsa walqamara husbaanaa; zaalika taqdeerul ‘Azeezil ‘Aleem
"[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)
Wa minan naasi wadda waaabbi wal an’aami mukhtalifun alwaanuhoo kazalik; innamaa yakhshal laaha min ‘ibaadihil ‘ulamaaa’; innal laaha ‘Azeezun Ghafoor
"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)
How can we benefit from knowing 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 a hunger or burning desire for success, you should seek it from the true possessor of might.
A good authentic supplication you can learn which incorporates a variety of Allah's names and attributes (including al-Aziz) is provided below. It was narrated that ‘Ubadah bin As-Samit said: “The Messenger of Allah (ﷺ) said: ‘Whoever wakes up in the morning and says upon waking: 
La ilaha illallah wahdahu la sharika lahu, lahul-mulku wa lahul-hamdu, wa Huwa ‘ala kulli shay’in Qadir; Subhan-Allah walhamdu lillahi, wa la ilaha illallahu, wa Allahu Akbar, wa la hawla wa la quwwata illa billahil-‘Aliyil-‘Azim
"None has the right to be worshipped but Allah alone, with no partner or associate. His is the dominion and all praise is to Him, and He is Able to do all things. Glory is to Allah, praise is to Allah, none has the right to be worshiped but Allah, Allah is the Most Great, and there is no power and no strength except with Allah, the Most High, the Most Supreme" The hadith then adds, "then he supplicates Rabbighfirli (O Lord, forgive me), he will be forgiven.'" Walid said: "Or he said: then if he supplicated, it will be answered for him then if he stood up and performed ablution and then performed prayer, his prayer would be accepted."