Although the phrase "Allahu Akbar" (الله أكبر) is a common phrase used by all Muslims in various situations, including the Salah (obligatory five prayers a day) and has even been used by some non-Muslims, as a show of support for the protesting Iranians; it is widely associated with the "radical" (fundamental) Muslims who shout it while engaged in Jihad - which usually entails the murder of innocent non-Muslims. Many 'moderates' and apologists claim it's simply the Arabic translation of a common English phrase meaning "God is great!"
However, this is untrue. "Allahu Akbar" does not mean "God is great" as claimed. It actually means "Allah is greater." Greater than what? You may ask. The fact that it's a 'war cry' for the Jihadis (more recently including Major Nidal Malik Hasan) should give you a clue to this. Let's examine the use of this phrase in scripture, and the use of these words in their original Arabic.
'Allah' is simply the Arab word for 'God.'
'God' in Arabic is translated 'ilah,' not 'Allah.'
Allah = الله (alif ا, lam ل, lam ل, ha ه)
ilah = اله (alif ا, lam ل, ha ه)
Transliteration: ašhadu ʾanla ilāha illal-Lāh, wa ʾašhadu ʾanna muḥammadan rasūlul-LāhLiteral: There is no god but Allah and Muhammad is the messenger of Allah.
As you can see, even though many English speaking Muslims falsely claim it states "there is no god but God", this is clearly not the case. Allah is not the generic word for 'god' in Arabic, but the name of Islam's deity - at the start of almost every Surah (Chapter) of the Qur'an is the phrase (in part): "In the name of Allah...."
A Treatise on Maliki Fiqh (Including commentary from ath-Thamr ad-Dani by al-Azhari)(310/922 - 386/996)
Why Muslims and Apologists insist 'Allah' means 'God'
For many (like Muhammad before them) a lie is worth being told if it will facilitate proselytisation (Da'wah). For others, 'moderate' Muslims and non-Muslims alike, it's purely out of ignorance. Language is always evolving, so the actual definition of a word and its popular usage can, and very often does, differ. For example, 'gay' still means 'happy' but through its use in western pop-culture, it can also mean 'homosexual'. This is also the case for the word "Allah" in predominantly Muslim cultures and societies. It is used interchangeably with 'God', as most people are Muslims; to them Allah is God, thus to make a distinction would be redundant.
It is unfortunate how so many are willing to take advantage of people's ignorance of the Arabic language when it suits their purpose. For example: the false conversion story of pop legend Michael Jackson. A video entitled 'Inshallah' (Allah willing) has been doing the rounds on user-contributed media sites like YouTube. It contains edited footage of Michael Jackson using the phrase 'Inshallah,' and its editor proudly proclaimed on its page "only Muslims say Inshallah." a view which was echoed by many of the Muslim commentators who viewed it. However, it was simply a case of selective editing. The Muslim who made the video had cut off the first part of Michael's statement. It was a 12 year-old promo for his Tunisian fans on the eve of his HIStory world tour. Since Arabic is Tunisia's official language and since Arabic-speaking Jews and Christians living in Tunisia also use the phrase 'Inshallah', it was not proof that he had converted to Islam.
In another case in Malaysia, the government banned Christians from using the word 'Allah' in reference to the Christian God, and in one incident, reported by CNN in October of 2009, twenty-thousand Bibles were seized by authorities because they referred to the Christian God as 'Allah,' completely disregarding the fact that due to the evolution of the Malay language, which has borrowed extensively from Arabic, Sanskrit and Portuguese, there is no indigenous Malay word for 'God' other than the pagan 'Allah'. It was reported that in 2010 a court ruling overturned the ban, a decision which the government has appealed against, insisting that it should remain in place. This decision to allow Christians to use the word 'Allah' has led to violent protests and bombings of several Malaysian churches. One of the protesters, named Faedzah Fuad, says:
So, according to Muslims, is 'Allah' the word for 'God' (i.e. the God of Abraham) or the name of their particular deity distinct from the God of Christianity and Judaism? No doubt you will receive very different answers depending on the situation. However, the original Arabic script found within the Qur'an tells you all that you need to know.
'Akbar' means 'great.'
'Akbar' in English is translated 'greater,' not 'great.'
كبير (kebir) = Great
أكبر (Akbar) = Greater
They are not interchangeable words. For evidence, we only need look to the Qur'an. Both kebir (great) and akbar (greater) are used in the following verse:
يسالونك عن الخمر والميسر قل فيها اثمتتفكرون كبير ومنافع للناس واثمهما اكبر من نفعهما ويسالونك ماذا ينفقون قل العفو كذلك يبين الله لكم الايات لعلكم
Yas-aloonaka AAani alkhamri waalmaysiriqul feehima ithmun kabeerun wamanafiAAu lilnnasiwa-ithmuhuma akbaru min nafAAihima wayas-aloonaka mathayunfiqoona quli alAAafwa kathalika yubayyinu Allahulakumu al-ayati laAAallakum tatafakkaroona
Yusuf Ali:They ask thee concerning wine and gambling. Say: "In them is great sin, and some profit, for men; but the sin is greater than the profit." They ask thee how much they are to spend; Say: "What is beyond your needs." Thus doth Allah Make clear to you His Signs: In order that ye may consider- 
Used in Context by Muhammad
One must also note that if the word "Allah" meant "God", why then would he be telling the Jews of Khaibar (who supposedly worship the same god) that Islam's Allah is greater?
Here are a few more quotes which use the phrase "Allahu Akbar".
Al-hamdu lillahi-lladhi hadana wa at amana wa saqana wa naamana. Allahu akbar. Allahumma'l fatna nimatik bi-kulli sharr. Fa asbahna minha wa amsayna bi-kulli khayr. Nasaluka tamamaha wa shukraha. La khayr illa khayruk. Wa la ilaha ghayruk. Ilaha'-saliheen wa rabba'l-alameen. Al-hamdu lillah. Wa la ilaha illa'llah. Ma sha'Allah. Wa la quwwata illa billah. Allahumma barik lana fima razaqtana. Waqina adhaba'n-na
'Allah' is not simply the Arabic word for 'God,' but the name of Islam's chosen deity (i.e. the pre-Islamic moon god located in the Ka'aba.) and 'Akbar' does not mean 'great,' but 'greater.' Greater than what? The answer is - Allah is greater...than whatever god you happen to believe in.
- Allahu Akbar - Muslim Arabic speaker, commenting on this page, confirms "Allahu Akbar" means "Allah is greater" (not "God is great)"
- "MALAYSIA: GOVERNMENT MAINTAINS ONLY MUSLIMS CAN USE ‘ALLAH’ TERM" - Compass Direct News, January 08, 2008
- Saeed Ahmed - "Bibles seized as Malaysia minorities fear fundamentalism" - CNN, October 29, 2009
- Mydans, Seth. "Churches Attacked in Malaysian ‘Allah’ Dispute", The New York Times, January 9, 2010.
- Quran root browser - ك ب ر - k-b-r