The people of Christianity have been taught to refer to their deity as "God." If you were to ask one of them: "What is your god's name?," they would respond "God!" (there are some exceptions). They object to Muslims worshipping "Allah," and usually picture "Allah" as some pagan god. Some of them will even go so far as to curse "Allah," not realizing that they are cursing "God."
Now the question becomes: where did the name "God" come from? Did Jesus (pbuh) ever say "God"? Did Moses (pbuh) ever say "God"? No! The Jews and Arabs are both Semitic tribes which descended from one father, Abraham (pbuh). Their languages are quite similar.
The Old Testament tells us that Moses (pbuh) referred to God as "El" or "Elohiym." Jesus (pbuh) too, referred to God using a similar construct. Jesus (pbuh) spoke Aramaic, however, the ancient copies of the Gospel available to us today are mostly written in Greek.
Very little of Jesus' actual words have been preserved to this day. However, we do know from Mark 15:34 that Jesus (pbuh) referred to God as "Eloi." "Eloi" is an Aramaic word which means "My God." It is pronounced as {el-o-ee'}. The Arabs would say the same word as "Elahi," pronounced {el-ah-ee'}. So Muslims refer to God with virtually the exact same word Jesus (pbuh) used.
Muslims are taught that Allah Almighty has more than one hundred names, the most well known among them being "Allah." These names are to be found in many places throughout the Qur'an. They embody the major characteristics of Allah Almighty such as "The Gracious," "The Merciful," "The Majestic," "The Supreme"...etc.. These names are usually considered adjectives, unless they are applied to Allah Himself, in which case they are treated as proper nouns. For instance:
"Allah's are the fairest names. Invoke Him by them. And leave the company of those who blaspheme His names. They will be requited what they do." The noble Qur'an, Al-Aaraf(7):180.
"Say (unto mankind): Supplicate unto Allah, or supplicate unto the 'Rahman' (Compassionate/Merciful/Gracious), unto whichever you supplicate (it is the same). His are the most beautiful names." The noble Qur'an, al-Isra(17):110.
"Allah! There is no god save Him. His are the most beautiful names." The noble Qur'an, Taha(20):8.
"Not equal are the Companions of the Fire and the Companions of the Garden: The companions of the Garden, they are the triumphant. Had We sent down this Qur'an on a mountain verily you would have seen it humble, rent asunder for fear of Allah. Such are the similitudes which We propound to humanity that they may reflect. He is Allah, other than whom there is no other god, He is the 'Knower' of (all things) both the unseen and the seen; He is the 'Gracious' the 'Merciful'. He is Allah, other than whom there is no god, the 'Sovereign' the 'Holy One' the (source of) 'Peace,' the 'Guardian of Faith' the 'Overseer,' the 'Majestic,' the 'Irresistible,' the 'Supreme': Glory be to Allah! (highly exalted is He) above the partners they attribute to Him. He is Allah the 'Creator,' the 'Innovator,' the 'Fashioner'. His are the Most Beautiful Names: Whatever is in the heavens and on earth do glorify Him: and He is the 'Mighty' the 'Wise'." The noble Qur'an, al-Hashir (59):20-24.