What Does ‘Subhana Rabbiyal Azim’ Mean?
Question

What is the meaning of the words “Subhana Rabbiy al-‘Azim (Glory be to my Lord, the Most Great)” and “Subhana Rabbiy al-A‘la (Glory be to my Lord, the Most High), that we say when bowing and prostrating, respectively? 
Is it obligatory to reflect upon the meaning when repeating these phrases, or is what is required is to reflect upon the majesty and perfection of Allah when bowing, and His exaltedness when prostrating, and so on?
Praise be to Allah.The meaning of tasbeeh
The meaning of tasbeeh (glorifying Allah) is declaring Allah, may He be exalted, to be far above any shortcoming or fault. So when you say “Subhan Allah (Glory be to Allah)”, what it means is: I declare You, O Lord, to be far above and free of any shortcoming or fault.
The meaning of al-‘Azim (the Most Great) is the One Who is possessed of ultimate greatness. The meaning of al-A‘la (the Most High) is the One Who is exalted in His Essence, and is exalted in His attributes.
Reflect upon the meaning of the Quran and adhkar during prayer
What is required of the worshipper is to reflect upon the meaning of what he recites of Quran and adhkar. In his recitation of Quran and dhikr he is required to bear in mind the meaning connected to these words, because every word that he utters in prayer has subtle wisdom and meanings, of which he will attain the goodness and benefits commensurate with his level of focus of mind and reflection on the meaning, and he will lose out on those benefits commensurate with the extent to which he is not able to focus and reflect on what he is reciting.
Reflecting on the great significance of the actions of the prayer
Bearing the meanings in mind is not limited only to the words of the prayer; rather it is also required to bear in mind the great significance of the actions of the prayer.
Shaykh Ibn ‘Uthaymin (may Allah have mercy on him) said: 
“One of the means that will help the worshipper not to let his mind wander whilst he is praying is to follow what he is saying and doing, and reflect upon the great wisdom behind the prescription of these words and actions. 
In the case of ruku‘ (bowing), for example, it is prescribed for the worshipper to glorify his Lord in his words and his actions. Hence the Prophet (blessings and peace of Allah be upon him) said: “…and in ruku‘, glorify the Lord.” 
Bowing before Allah is glorifying Him by one’s actions, and saying “Subhana Rabbiy al-‘Azim (Glory be to my Lord the Most Great)” is glorifying Him by one’s words. All that is left is for the worshipper to glorify Him in his heart, and this can only be attained with presence of mind. Hence in ruku‘, there is glorification of Allah in one’s words and actions, and also in one’s heart.” (Fatawa Nur ‘ala ad-Darb, 8/2)
Ibn Rajab said: 
“When the worshipper shows humility to his Lord by bowing and prostrating, he is attributing to his Lord the attributes of might, majesty, greatness and exaltedness. It is as if he is saying: Humility and humbleness are my attributes, and exaltedness, greatness and majesty are Your attributes. Hence it is prescribed for the worshipper when bowing to say: “Subhan Rabbiy al-‘Azim (Glory be to my Lord the Most Great)” and when prostrating to say: “Subhan Rabbiy al-A‘la (Glory be to my Lord Most High).” The Prophet (blessings and peace of Allah be upon him) sometimes used to say when bowing and prostrating: “Subhana Dhi’l-jabaruti wa’l-malakuti wa’l-kibriya wa’l-‘azamah (Glory be to Him Who possesses might, sovereignty, majesty and greatness).” (Al-Khushu‘ fi’s-Salah, pp. 41-43)
How to focus on prayer
The questioner says: Is it obligatory to reflect upon the meaning when repeating these phrases, or is it that what is required is to reflect upon the majesty and perfection of Allah when bowing, and His exaltedness?
The answer is that what is required of the worshipper is to reflect upon the meaning of what he is saying of Quran and dhikr , and on what he is doing of actions, for bowing and prostration are prescribed for the glorification of Allah, may He be exalted, and the prescribed dhikr [in prayer] has the meaning of glorification and veneration.
Based on that, whoever reflects on the meanings of what he says of adhkar when bowing and prostrating will inevitably lead him to reflect upon the might and majesty of Allah, may He be exalted. But his reflection should be commensurate with what he is saying and doing in the prayer, and he should not overdo it, for that is waswas (whispers) from the Shaytan, aimed at distracting him from his prayer.
Al-Ghazali said in Ihya ‘Ulum ad-Din (1/150): 
“You should understand that one of the tricks [of the Shaytan ] is to distract you in your prayer by making you remember the Hereafter and think of what good deeds you could do, so as to prevent you from focusing on what you are reciting. Therefore, you should realise that anything that distracts you from focusing on the meanings of what you are reciting comes under the heading of waswas.”
And Allah knows best.