(The Magnificent One)
The Great One, The Mighty, The Tremendous.
Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Azeem, The Supreme Glory and The Greatest. His reality is too great to be fully understood under the human mind's capacity. He is infinitely greater than His creation. It is said, His throne (symbolic of His knowledge and authority) extends over the heavens and the earth, and their care burdens Him not. He is the One deserving the attributes of Exaltment, Glory, Extolement, and Purity from all imperfection.
From the root ayn-za-mim (ع ظ م), which has the following classical Arabic connotations: to be great, large, big, to be vast, huge, enormous, to be great in rank, importance or dignity, to be imposing, powerful, superior, to be magnificent, honored, revered, to be immense, stupendous, lofty, to be above imperfection and sublime.
Understanding Azeem as it is mentioned in the Qur'an:
When it comes to Al-Azeem (Al-Azim), we need to understand that with Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ there are no limits or restrictions. He is tremendous in all aspects. His mercy is tremendous, His ability, His strength, His knowledge, His generosity are all as great as can be. He is truly the possessor of excellence. He is Al-Azeem, The Greatest in all His beautiful qualities and attributes. This name is mentioned nine times in the Qur'an but is also used to describe things more generally as Azeem (tremendous). For example, His punishment in the hereafter is described as azeem. The Qur'an (15:87), the throne (27:26, 27:23), His grace (33:35), the day of judgment (83:5), the trials and tribulations of the children of Israel (7:141), His rewards (64:15), the upright character of Prophet Muhammad (ﷺ) (68:4) are all referred to as azeem.
Fasabbih bismi rabbikal azeem
"So exalt the name of your Lord, the Most Great." — (Qur'an 55:96) & (Qur'an 69:52)
Connecting with this name in your Salah:
Be mindful of this name when you go into Ruku and recite, سُبْحَانَ رَبِّيَ الْعَظِيمِ subhana rabbiyal azeem meaning, "Glorious is my Lord the Magnificent."  Think of the words you're saying as you recite them and understand the implications behind them. Why do you offer prayer? What is it your attending to? We want Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ to forgive us, to give us the best life in this world and in the next, to ease our tasks and make us better Muslims for Al-Azeem is tremendous and all capable.
Allahu laaa ilaaha illaa Huwal Haiyul Qaiyoom; laa taakhuzuhoo sinatunw wa laa nawm; lahoo maa fissamaawaati wa maa fil ard; man zal lazee yashfa’u indahooo illaa bi-iznih; ya’lamu maa baina aydeehim wa maa khalfahum wa laa yuheetoona bishai’im min ‘ilmihee illaa bimaa shaaa’; wasi’a Kursiyyuhus samaawaati wal arda wa laa Ya’ooduhoo hifzuhumaa; wa Huwal Aliyyul ‘Azeem
"Allah – there is no deity except Him, the Ever-Living, the Sustainer of [all] existence. Neither drowsiness overtakes Him nor sleep. To Him belongs whatever is in the heavens and whatever is on the earth. Who is it that can intercede with Him except by His permission? He knows what is [presently] before them and what will be after them, and they encompass not a thing of His knowledge except for what He wills. His Kursi extends over the heavens and the earth, and their preservation tires Him not. And He is the Most High, the Most Great." — (Qur'an 2:255)
Lahoo maa fis samaa waati wa maa fil ardi wa Huwal ‘Aliyul ‘Azeem
"To Him belongs whatever is in the heavens and whatever is in the earth, and He is the Most High, the Most Great." — (Qur'an 42:4)
A believer benefits in knowing that Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ is Al-Azeem by calling upon this name whenever faced with a problem. There is no obstacle too large, no dream too big that Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ cannot fulfill. He can make any aspect of your life great. Even in the mundane, He can show you the beauty in it. The Prophet ﷺ would make this du'a when distressed which calls upon two of Allah's attributes of being Al-Azim and Al-Haleem (The Clement One).
Narrated Ibn Abbas: The Prophet ﷺ used to invoke Allah at the time of distress, saying, 
La ilaha illal-lahu Al-`Azim, al- Halim, La ilaha illal-lahu Rabbu-s-samawati wal-ard wa Rabbu-l- arsh il-azim.
"There is no God but Allah, the Magnificent, the Forbearing. There is no God but Allah, Lord of the magnificent Throne. There is no God but Allah, Lord of the heavens and Earth, Lord of the generous Throne."
How to tip the scale in your favor with the name Al-Azim: The Prophet (ﷺ) said, "(There are) two words which are dear to the Beneficent (Allah) and very light (easy) for the tongue (to say), but very heavy in weight in the balance. They are: 'Subhan Allah wa-bi hamdihi' and 'Subhan Allah Al-Azim." 
This name also brings about humility in all who understand this attribute of Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ. As mentioned previously, a constant theme in many of these names is that we're deluded in worldly pursuits. A person who sees success can think of himself as being grandeur. But in reality, nothing can eclipse Allah's greatness. We can think of ourselves as big fish, not realizing we're swimming in a little pond.
In another case, we unknowingly may be influenced by a person with high status. Many fall into a trance of some sort, bending over backward to appease their demands. If, at that moment, Al-Azeem popped into your mind, it takes away the power they may hold over you. We realize yes, they may be wealthy, famous, and influential, but in comparison to Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ they are nothing. Therefore, you don't need their approval. The only support we need, the only one that matters and can genuinely help us, is Allah سُبْحَٰنَهُۥ وَتَعَٰلَىٰ.
Another authentic supplication we can learn from the hadith which incorporates a variety of Allah's names and attributes. 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."
 Sahih (Darussalam) Jami At-Tirmidhi 262
 Sahih al-Bukhari 6345 & Sahih Muslim 2730
 Sahih al-Bukhari 7563
 Sahih (Darussalam) Sunan Ibn Majah 3878