Short answer
The name of God in Islam is Allah. It is the Arabic word for the one true God, and Arab Christians use it too. Allah also has many beautiful names, such as ar-Rahman (the Most Merciful), that describe Him.
Allah is the proper name of the one God, and many Sunni scholars call it His greatest name, because all His other names are attributed to it. His names are fixed by revelation (tawqifi): Muslims call Him only by names found in the Quran and Sunnah.
The four schools agree on this.
Explanation
"Allah" is the personal name of God in Arabic. Many scholars say it is not derived from any other word, and it has no plural and no gender, which suits a God who is One. Other grammarians trace it to al-ilah, "the God". Either way it does not mean a moon god or a tribal idol, a claim that appears online with no basis in Arabic or in the Quran.
Muslims are not the only ones who use it. Arabic Bibles use the word Allah for God, and Arab Christians pray to Allah in church. What differs between the faiths is not the word but what is believed about Him. Muslims hold that He is one, with no son and no partner.
Besides this name, Allah has many others, called al-asma al-husna (the most beautiful names): ar-Rahman (the Most Merciful), al-Ghafur (the Forgiving), al-Alim (the All-Knowing) and more. The Quran says to Him belong the best names, so call on Him by them. These names come from revelation, so Muslims do not invent new names for Allah, even ones that sound respectful. Calling Him by a description is different: saying "the Creator" or "my Lord" is fine.
In English, saying "God" for Allah is correct, and many Muslims do so. What matters is that the One you mean is the God the Quran describes. You can learn each name with its meaning on our page of the 99 names of Allah.
The evidence
Quran, Surah Taa-Haa 20:8
ٱللَّهُ لَآ إِلَٰهَ إِلَّا هُوَ ۖ لَهُ ٱلْأَسْمَآءُ ٱلْحُسْنَىٰ ﴿٨﴾
Allah - there is no deity except Him. To Him belong the best names.
Names Allah as the only deity and says the best names belong to Him.
Quran, Surah Al-Israa 17:110
قُلِ ٱدْعُوا۟ ٱللَّهَ أَوِ ٱدْعُوا۟ ٱلرَّحْمَٰنَ ۖ أَيًّۭا مَّا تَدْعُوا۟ فَلَهُ ٱلْأَسْمَآءُ ٱلْحُسْنَىٰ ۚ وَلَا تَجْهَرْ بِصَلَاتِكَ وَلَا تُخَافِتْ بِهَا وَٱبْتَغِ بَيْنَ ذَٰلِكَ سَبِيلًۭا ﴿١١٠﴾
Say, "Call upon Allah or call upon the Most Merciful. Whichever [name] you call - to Him belong the best names." And do not recite [too] loudly in your prayer or [too] quietly but seek between that an [intermediate] way.
Shows that Allah may be called by any of His names, such as ar-Rahman.
Sahih al-Bukhari 2736
Sahih... إِنَّ لِلَّهِ تِسْعَةً وَتِسْعِينَ اسْمًا مِائَةً إِلاَّ وَاحِدًا ...
"Allah has ninety-nine names, one hundred less one; whoever learns and keeps them will enter Paradise."
Confirms that Allah has many names besides the name Allah, and that learning them is rewarded.
Where the ruling is recorded
- Sharh al-Aqa'id al-Nasafiyya (al-Taftazani), Section on the names and attributes of Allah. Explains that the names of Allah are taken from revelation (tawqifi).
- al-Fiqh al-Akbar (Imam Abu Hanifa), Section on the names and attributes. Affirms the names and attributes of Allah as eternal and unlike those of creation.
This answer is pending review by the scholars on our team. Quran translation by Sahih International. Hadith Arabic from the standard editions of the six books and the Muwatta, with gradings as recorded there; the English is our own rendering. Spotted a mistake? Tell us.