proof of god
the necessary existent
Ibn Sina's proof of God, on its most basic level, boils down to cause and effect. Ibn Sina posits that everything in the universe exists because it was caused by something else. This is a notion that tends to be accepted in the modern, scientific community and many fields of science are dedicated to finding causation for the phenomena we experience. With any physical object, we can create a chain of "causes" that lead to its existence. For example, the computer that I'm typing on exists because it was put together in a factory in China. It was put together because a company in California paid for the machine to be put together. The company paid for the manufacturing because it saw profit in creating and selling computers. The company saw profit because people were showing interest in owning computers for personal use. People showed interest in owning computers because of its usefulness as a tool. However, each feature we find useful in a computer has a similar train of causality, as does the plastic, silicone, aluminum, and other materials used to create the physical computer.
As you can see, after only a few layers of extrapolation, the explanations become very vast and very abstract. And that's exactly what ibn Sina concerns himself with. Ibn Sina noticed a pattern that each link in the chain comes from an external causality. For example, the computer cannot create itself; something outside of the computer has to create it. In the same vein: a person cannot create him or herself, they're created by two other people: their parents. Ibn Sina noticed this pattern and then skipped to the end of the chain of causality. He argued that if everything is created by an external entity, then the universe itself must be created by an external entity. Ibn Sina called this external entity the "Necessary Existent" because its existence is necessary for the existence of every other thing. The Necessary Existent is the force (or in Ibn Sina's case: the being) that created the first cause, which then proliferated in all the other causes that make up the universe in which we live today.
attributes of god
For ibn Sina, the Necessary Existent is the God of Abrahamic religions (and, as some of his contemporaries argue, the highest order God of all other theistic religions). After establishing God's existence through cause and effect, ibn Sina begins to describe what attributes would befit a being in such a precarious position. In his analysis, ibn Sina cross-references the Quran, and the attributes it ascribes to God. For this summary, I'll bring in verses from Surah Ikhlas, which is also found in the sixth part of the Ismaili du'a. Initially, ibn Sina's God cannot, by definition, have any cause; if it did have a cause, it wouldn't be the Necessary Existent and ibn Sina would have to extrapolate further to find God. This relates to verse three of Surah Ikhlas: "he was not born," meaning that God did not come from a cause.
Secondly, God cannot be made of constituent parts. As humans, we are made of organs which allow us to function. Those organs are made of cells which allow them to function. Those cells are made of proteins, those proteins out of molecules, et cetera and so forth. Each of these smaller levels can be seen as a "cause" of the levels above it. We as humans wouldn't be able to exist if carbon-14 (for example) didn't exist. Because the Necessary Existent doesn't have any causal factors, it cannot have constituent parts. We can relate this to the first verse of Surah Ikhlas: "God is one," meaning that God is made up of one piece, not individual pieces coming together.
Ibn Sina makes further arguments that all the attributes ascribed to God in the Quran (which you can find in the 99 Names of God) must apply to the Necessary Existant. However, it's here that his argument becomes less stable. Because all the attributes (just the rest of the Universe) come from God, he can be said to embody them, which is what ibn Sina purports. However, God by this definition isn't limited to the positive attributes associated with him. Because God is all-encompassing, he also holds the opposite of his attributes, simultaneously. For example, if God is described as The Giver of Life (المحيي), He must also be the Bringer of Death (المميت). If He is The Maker (البارئ), He must also be the destroyer. This can easily tread into uncertain territory with attributes such as The Praiseworthy (الحميد) and The Truth (الحقّ). However, according to ibn Sina, both truth and falsehood came from the Necessary Existent.
further learning
Ibn Sina created a thought experiment to further prove the existence of God. This is covered on the Falling Man page.
About two hundred years later, Thomas Aquinas expanded on ibnSina's hypothesis. This video from the wonderful people at Crash Course covers Aquinas' conclusion: