The name Allah (الله) is the Arabic term for the God of Islam. This theonym is likely a contraction of Arabic Al-'ilāh (ال إِله), which means "the god". It is thus related to the Hebrew term Eloh(-im).
In Islam, Allah is the personal name of God and comprises all of the 99 Names of God in Islam. When a Muslim refers to Allah, he thinks of God as being One, Creator, Owner, Sustainer, and Almighty. In Arabic, the name is composed of four letters, ا ل ل ه (or Alif, Lam, Lam, Ha) which when brought together make الله. Arab Christians often incorrectly use this name simply as a generic term "god". (The actual generic word for "god" is simply 'ilāh, though due to the strong association of Arabic with Islam this word is rarely used in modern speech.) These Christians do maintain a belief in the traditional Christian Trinity, however, and believe that Allah consists of the Father, Son and Spirit.
Christianity and Judaism
As noted, the name Allah is used by Arab Christians throughout the Middle East. It is also used by Arabic Jews, especially those residing in Yemen. According to the Qur'an the authority of the Torah and the Gospels is subordinated to the content of Muslim revelation. It was regarded by the prophet Muhammad that both, the Jewish and Christian scriptures, were "corrupted" (from the Arabic term: Tahrif). Accordingly, since the Qur'an has remained unaltered to the present day, (compared to the other Scriptures) Muslims assert that only the Qur'an contains the Words of God -- the truth in toto (i.e., the whole truth and nothing but the truth). Whereas the earlier Scriptures/Books such as The New Testament, The Old Testament, etc. contain only partial truths. 
Christians believe in the mystery of the Trinity (Father, Son, and Holy Spirit), but Muslims don't. In the Qur'an's Surah 13: "Allah is the Creator of all things: He is the One, the Supreme and Irresistible."
The Kalimah, the first of the 5 pillars of Islam, states:
To believe that Allah Ta’ala is One. That He has no partner, no father, mother, wife and so has no son or daughter; and that there is nothing else that is worthy of praise, or worship except Him ‑Alone and to equally testify that Muhammed (S.A.W.) is His servant and messenger.
This Kalimah is simple and can be uttered in a few sentences which can easily be pronounced by anybody without much difficulty.
"La ilaha Illa Allah Muhammad Rasulu llah" (SAW).
Meaning: There is no deity worthy of worship except Allah and that Muhammad (S.A.W.) is the messenger of Allah. 
Consequently Allah is unique, is the God in Islam, and according to Marshall G. S. Hodgson, (The Venture of Islam) Muslims are worshiping a Allah rather than God, the creator.
From the 112th sura of the Qur'an: "In the name of God, the Merciful, the Compassionate. Say (O Mohammad) He is God the One God, the Everlasting Refuge, who has not begotten, nor has been begotten, and equal to Him is not anyone." 
Possible Early Use
It is a matter of contention as to whether the name Allah came into use only with the foundation of Islam, or whether it is a reuse of a title of the main pagan god of pre-Islamic Mecca; Hubal, god of the Moon. Muslim scholars have offered reasonable refutations of this allegation, but some Christian groups have offered counter-refutations that are seemingly logical as well. Whether the allegation is true is unknown.
Takbeer (Glorification of God)
Takbeer (تَكْبِير) (Glorification of God), is the common way Muslims give glory to God. The wording used is Allahu-Akbar "الله أكبر", which is translated as "God is greater". It is a very common practice in Islam and is essential to the Islamic call of prayer (Athan).