In a nutshell:
Ilaah (إله) is the Arabic term for God. It refers to divine beings who people worship. In the Islamic context, it refers to Allah, The One and Only God Who deserves to be worshiped.
Context
The term Ilaah is used to refer to any god worshiped and loved by people. as Allah says:
وَقَالَ الْمَلأُ مِنْ قَوْمِ فِرْعَوْنَ أَتَذَرُ مُوسَى وَقَوْمَهُ لِيُفْسِدُوا فِي الأَرْضِ وَيَذَرَكَ وَآلِهَتَكَ
And the chiefs of pharaoh's people said, `Wilt thou leave Moses and his people to create disorder in the land and forsake thee and thy gods (ilaahs)?' (Surat al-'A'raf 7: 127)
Ibn Abass commented on "forsake thee and thy gods (Ilaahs)?" saying:
وَيَذَرُكَ وَآلِهََتُكَ قَالَ : يَتْرُكُ عِبَادَتُكَ
"they will leave you and your worship" (Suyuti, ad-Dur al-Manthur, Vol. 3, p. 516)
The term in the Islamic context refers to the One and Only God Who deserves to be worshiped. It is mentioned in the testimony of Islam: There is no god (Ilaah) but Allah.
Al-Raghib al-Asfahanee (d.425) said ilaah is a name given to every object that is served:
فَاْلإِلَهُ عَلَى هَذَا هُوَ اْلَمَعْبُودُ
"Ilaah as such is the Worsheped (Being)" (Asfahani, Mufradat al-Quran, p. 21)
Ibn Taymyah commented on this and said:
هُوَ الْمَعْبُودُ اْلمُطَاعُُ؛ ... فَإِنَّ اْلإِلَهُ هُوَ المَأْلُوهُ، وَالمَأْلُوهُ الَّذِي َيسْتَحِقُّ أَنْ يُعْبَدَ
"He is the worshed, obeyed ... the Ilaah is the one who deserves to be worshiped." (Ibn Taymyah, MAjmu' al-Fatawa, Vol. 5, p. 227)
Ibn Mandhoor (d. 711) defined it as "any object that is taken as an object of servitude/obedience" (Lisaan al-Arab).
Conclusion
Ilaah (إله) is god that people worshiped, but Islam denounced the notion of worshiping others besides Allah, so Allah is the Only (Ilaah) God Who deserves to be worshiped.
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