Praise be to Allaah. Firstly:  The words of tasbeeh, “Subhaan Allaah”, incorporate one of the greatest fundamental meanings of Tawheed and one of the basic principles of belief in Allaah, may He be glorified and exalted, which is declaring Him to be above any fault, shortcoming, corrupt notions or false idea.   The basic linguistic meaning points to this and is taken from the word al-sabh which means distance or being far from something.  Al-‘Allaamah Ibn Faaris said: The Arabs say: Subhaana min kadha meaning How far away is such and such. Al-A’sha said:  Subhaana min ‘Alqamat al-faakhir aqoolu lamma jaa’ani fakhruhu How surprising is ‘Alqamah when he boasts; I said that when I heard about his boasting.  Some people interpreted these lines as meaning: It is very surprising when ‘Alqamah boasts, and this is close to the meaning of distance, because the speaker is far away from pride. End quote.  Mu’jam Maqaayees al-Lughah (3/96).  So tasbeeh of Allaah means declaring that one’s heart and mind are far removed from thinking that there is any shortcoming in Him, or attributing any evil to Him, and declaring Him to be far above any fault that is attributed to Him by the mushrikeen and heretics.  It is this meaning that is indicated in the Qur’aan.  Allaah says (interpretation of the meaning):  “No son (or offspring) did Allaah beget, nor is there any ilâh (god) along with Him. (If there had been many gods), then each god would have taken away what he had created, and some would have tried to overcome others! Glorified be Allaah [Subhaan Allaah] above all that they attribute to Him!” [al-Mu’minoon 23:91]  “And they have invented a kinship between Him and the jinn, but the jinn know well that they have indeed to appear (before Him) (i.e. they will be called to account). 159. Glorified be Allaah [Subhaan Allaah]! (He is free) from what they attribute unto Him!” [al-Saffaat 37:158-159]  “He is Allaah, beside Whom Laa ilaaha illa Huwa (none has the right to be worshipped but He), the King, the Holy, the One Free from all defects, the Giver of security, the Watcher over His creatures, the All‑Mighty, the Compeller, the Supreme. Glory be to Allaah! (High is He) above all that they associate as partners with Him” [al-Hashr 59:23]  It also appears in the report narrated by Imam Ahmad in al-Musnad (5384) from Hudhayfah (may Allaah be pleased with him), describing the recitation of the Prophet (peace and blessings of Allaah be upon him) in the night prayer. He said: “When he came to a verse which spoke of Allaah being above any fault or defect, he said Subhaan Allaah.” Classed as saheeh by al-Albaani in Saheeh al-Jaami’ (4782) and by the editors of al-Musnad.  Imam al-Tabaraani narrated in his book al-Du’aa’ a number of reports explaining this meaning of this word, which he compiled in a chapter entitled Tafseer al-Tasbeeh (explanation of the phrase Subhaan Allaah) (p. 498-500). Among the reports that he quoted in this chapter are the following:  It was narrated from Ibn ‘Abbaas (may Allaah be pleased with him):  Subhaan Allaah means declaring Allaah to be above every bad thing.  It was narrated that Yazeed ibn al-Asamm said:  A man came to Ibn ‘Abbaas (may Allaah be pleased with him) and said:  “Laa ilaaha ill-Allaah” we know: there is no god other than Him. “Al-hamdu Lillaah” we know: all blessings come from Him and He is the One Who to be praised for them. “Allaahu akbar” we know: there is nothing greater than Him. But what does “Subhaan Allaah” mean?  He said: This is a word that Allaah has approved for Himself, and has enjoined His angels to say, and inspired the elite of His creation to utter.  It was narrated that ‘Abd-Allaah ibn Buraydah narrated that a man asked ‘Ali (may Allaah be pleased with him) about the phrase “Subhaan Allaah” and he said: It is veneration of the majesty of Allaah.  It was narrated that Mujaahid said:  Tasbeeh means to declare that Allaah is free of anything bad.  It was narrated that Maymoon ibn Mahraam said:  Subhaan Allaah is veneration of Allaah and is a phrase by which Allaah is venerated.   It was narrated that al-Hasan said:  Subhaan Allaah is a restricted phrase that cannot be used with reference to any of creation.  It was narrated that Abu ‘Ubaydah Mu’ammar ibn al-Mathanna said:  Subhaan Allaah is declaring Allaah to be above (all defects).  Al-Tabaraani said: al-Fadl ibn al-Habbaab told us: I heard Ibn ‘Aa’ishah say:  When the Arabs find something objectionable and regard it as very serius, they say “Subhaan”. It is as if it is declaring Allaah to be above every bad thing that it is inappropriate to attribute to Him.  Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said:  The command to glorify Him (by saying Subhaan Allaah) also implies declaring Him to be above every fault and shortcoming, and affirming the attributes of perfection for Him. Tasbeeh implies declaring Him to be above any fault and venerating Him, and veneration implies affirming praiseworthy attributes for which He is praised, and that implies declaring Him to be above any faults, praising Him, magnifying Him and affirming His Oneness. End quote.  Majmoo’ al-Fataawa 16/125  Secondly:  With regard to the meaning of the phrase bi hamdihi: in brief, it means combining tasbeeh (glorification) and hamd (praise), either simultaneously or consecutively. So the meaning is: I glorify Allaah as I am praising Him or I glorify Allaah and I praise Him.  Al-Haafiz Ibn Hajar (may Allaah have mercy on him) said:  With regard to the words “wa bi hamdihi”:  It was said that the waw indicates simultaneous action, so the meaning is: I glorify Allaah at the time when I am praising Him for His help and support. And it was said that it refers to consecutive actions, so the meaning is: I glorify Allaah and I praise Him.  It may be that the meaning is: I laud Him by praising Him. So “Subhaan Allaah” is a separate phrase, and “bi hamdihi” is separate phrase.  Al-Khattaabi said concerning the phrase “Subhaanaka Allaahumma Rabbana wa bi hamdika (Glory be to You, O Allaah, our Lord, and praise) – i.e., by means of Your strength which is a blessing for which You are to be praised I glorify You, not by my power and strength. End quote.  Fath al-Baari (13/541). See also al-Nihaayah fi Ghareeb al-Hadeeth by Ibn al-Atheer (1/457).  Thirdly:  With regard to the meaning of the Basmalah, “Bismillaah (in the name of Allaah)”, this has been explained and discussed in the answer to question no. 21722.  And Allaah knows best. http://www.islamqa.com/en/ref/104047