I’m not a Muslim. But I’m curious as to what Muslims believe is the answer to this question. Do all Muslims agree on one definition? What do you think? Who is God?
Yes, the God has been described well in Qur'an. But the descriptions are not like mathematical proofs but are philosophical and infrastructural; Here are the examples from Qur'an:
- They say who can make a death alive again; say who have been created them first time from nothing! [36:78,79] [76:1]
- Every object will be destroyed except God! [28:88]
- God is the most merciful; but also the most punisher according to his selection! [40:3] [3:129]
- God provides help and safety and is not under anything's help or provided safety [23:88]
- God is alive forever [25:58]
- When God wants a thing to happen, he just says: "be" and that thing will be [2:117]
- The God is who have been created the night and day traffic and who are policing the universe [23:80] [2:255]
- The God is Begin and is End! [57:3]
- The God is who have been hidden daeth time to recompensed everyone according to that for which it strives! [20:15]
- The God is who has in his hands the un-physical aspects of objects! [36:83]
- The God is who which there is no any other God except him. [2:255]
- The God is who that keeping the universe does not tire he and sleepy never will get him [2:255]
- And he is who keeps previous very past human's information and does not forget. [20:52]
Imam Reza (A.S) (8th Shia Imam) also has an excellent description of God:
He says: [Shaykh al-Saduq, al-Tohid, page ~480]
- Is not in minds rather than forgiven!
- Is not known rather than unknown!
- Make changes on objects and himself does not change!
- Sees eyes while eyes can not see him!
Hope these help :)
Sūrat al-Ikhlāṣ provides a good insight about how Muslims view Allah:
 قُلۡ هُوَ اللّٰہُ اَحَدٌ
 اَللّٰہُ الصَّمَدُ
 لَمۡ یَلِدۡۙ وَ لَمۡ یُوۡلَدۡ
 وَ لَمۡ یَکُنۡ لَّہٗ کُفُوًا اَحَدٌ
Say: He is Allah, the One!  Allah, the eternally Besought of all!  He begetteth not nor is he begotten.  And there is none comparable unto Him. 
- Allah is the One and only (true) god.
- He created everything and He Himself was not created.
- He has no progeny nor is He a progeny of another.
- There are none like Him (in power or form).
Also, the 99 Names of Allah are another source of His attributes. They are far too many to list here, but this Wikipedia article does a good job of listing them with English translations: http://en.wikipedia.org/wiki/99_Names_of_God
Here, the consensus ends. Some Shi'a scholars believe that Allah has no physical form and cannot be seen (by the eye). Among Sufis there is a belief that the whole universe is a manifestation of Allah. Some Sunnis, on the other hand, believe that Allah does have a physical presence that is vaguely human (with hands and feet). Many others (from all sects) believe that this is an unanswerable question.
In Islam, God is the creator of the Universe. He is the the most powerful being, the most most beneficent, the most merciful. There are in infact 99 names for God in Islam. All these names depict a quality in God. In other words, they are praise worth names of God.
There is no conflict among the definition of God in Islam, what so ever. In fact there is one chapter in Quran about who is God, named Tashid. The translation is
(1) Say: He, Allah, is One. (2) Allah is He on Whom all depend. (3) He begets not, nor is He begotten. (4) And none is like Him.
In Chapter Yasin, the power of God is depicted as
Is not He, Who created the heavens and the earth Able to create the like of them? Yes, indeed! He is the All-Knowing Supreme Creator.
Verily, His Command, when He intends a thing, is only that He says to it, "Be!" and it is!
This basically means, when he wishes to do something, it happens automatically.
Also the main crux of Islam is to recognize God and say that he is the only God and there is no one beside him who is worthy of worship.
The second clause in Islam is to recognize Prophet Muhammad who was the true prophet of God through him we know God.
Talking from a Shia perspective, @YasserZamani and @Ahmadi have answered the question very good, and from the other answers you can easily see that not all the Muslims agree on one understanding of Allah. However, let me put the issue differently so that you can hopefully find more coherence behind all of them.
All those who believe in a Heavenly religion, Muslims, Christians and Jews, as far as I know define God as a perfect being. This very beginning is the same in all such perspectives. However, the interpretation of being perfect is somewhat different and this is understandable from the distinctions between teachings of different schools and religions.
For example, God is considered as omnipresent and omnipotent in all such schools and religions but then see how differently they would use these reputations to talk about their God, which is of course a same God for all of them (different understanding of a same notion):
Christians (at least a good number of them) say God is omnipotent, so He CAN do whatever that He wills, and begetting a man as His son or incarnating into a flesh being two examples for that, see e.g. this question. Also a number of Christians think God being omnipresent means the God exists everywhere inside and outside of His created universe (or even universes with any possible number of dimensions), see e.g. this question.
Sunni Muslims (at least a good number of them) say similar sayings but also with some differences. They also believe God being omnipotent allows Allah to be visible in the Day to the true believers, with the very material eyes on face that we do posses now, see e.g. this question. Also some Sunni Muslims believe in Allah having body and from this respect they are again thinking like the Christians and maybe Jews, although the body will not be like a man as the Christians think about Jesus --peace be upon him. Also they (with the stress over the fact that not all of them) assume God to be everywhere in the sense a Christian may assume. Even they sometimes specify God's position and then talks about the movement of God, see e.g. this question or this one.
Shia Muslims say completely differently, the God is neither visible in this life nor in the Day or the Hereafter by any material eye, but only with the eyes of heart, e.g. see this question. The God has no body and no spirit but is pure of anything that He Himself has created. He also is nowhere in the sense that He is anywhere or fills the space. He has no time and no spatial dimension, even neither infinity. All such characteristics is proved in Shia theology to introduce limitations (imperfections) to a being and so the God, the perfect, would be free from all such qualities.
So you see, from the very fundamental levels of teachings of the prophets and their successors --peace be upon them all-- all of us, Shia or Sunni, Muslim or Christian or Jew, think the same, God is perfect, but then some have their own interpretations of those teachings and one should personally seek for the one interpretation which appear as rigorous and reasonable:
وَنَزَعْنَا مِن كُلِّ أُمَّةٍ شَهِيدًا فَقُلْنَا هَاتُوا بُرْهَانَكُمْ فَعَلِمُوا أَنَّ الْحَقَّ لِلَّـهِ وَضَلَّ عَنْهُم مَّا كَانُوا يَفْتَرُونَ
And from each people shall We draw a witness, and We shall say: "Produce your Proof": then shall they know that the Truth is in Allah (alone), and the (lies) which they invented will leave them in lurch. [28:75]
أَلَمْ تَرَ إِلَى الَّذِينَ يُزَكُّونَ أَنفُسَهُم ۚ بَلِ اللَّـهُ يُزَكِّي مَن يَشَاءُ وَلَا يُظْلَمُونَ فَتِيلًا \ انظُرْ كَيْفَ يَفْتَرُونَ عَلَى اللَّـهِ الْكَذِبَ ۖ وَكَفَىٰ بِهِ إِثْمًا مُّبِينًا \ أَلَمْ تَرَ إِلَى الَّذِينَ أُوتُوا نَصِيبًا مِّنَ الْكِتَابِ يُؤْمِنُونَ بِالْجِبْتِ وَالطَّاغُوتِ وَيَقُولُونَ لِلَّذِينَ كَفَرُوا هَـٰؤُلَاءِ أَهْدَىٰ مِنَ الَّذِينَ آمَنُوا سَبِيلًا
Hast thou not turned Thy vision to those who claim sanctity for themselves? Nay-but Allah Doth sanctify whom He pleaseth. But never will they fail to receive justice in the least little thing. / Behold! how they invent a lie against Allah! but that by itself is a manifest sin! / Hast thou not turned Thy vision to those who were given a portion of the Book? they believe in sorcery and Evil, and say to the Unbelievers that they are better guided in the (right) way Than the believers! [4:49,50,51]