This Attribute may be the most easy to comprehend. God is not a composite Being. The Church tells us that our soul is "simple". Aquinas says it best, in my opinion.
Summa Theologica I;3 http://www.newadvent.org/summa/1003.htm
On the contrary, Augustine says (De Trin. iv, 6,7): "God is truly and absolutely simple."
I answer that, The absolute simplicity of God may be shown in many ways.
First, from the previous articles of this question. For there is neither composition of quantitative parts in God, since He is not a body; nor composition of matter and form; nor does His nature differ from His "suppositum"; nor His essence from His existence; neither is there in Him composition of genus and difference, nor of subject and accident. Therefore, it is clear that God is nowise composite, but is altogether simple.
Secondly, because every composite is posterior to its component parts, and is dependent on them; but God is the first being, as shown above (Question 2, Article 3).
Thirdly, because every composite has a cause, for things in themselves different cannot unite unless something causes them to unite. But God is uncaused, as shown above (Question 2, Article 3), since He is the first efficient cause.
Fourthly, because in every composite there must be potentiality and actuality; but this does not apply to God; for either one of the parts actuates another, or at least all the parts are potential to the whole.
Fifthly, because nothing composite can be predicated of any single one of its parts. And this is evident in a whole made up of dissimilar parts; for no part of a man is a man, nor any of the parts of the foot, a foot. But in wholes made up of similar parts, although something which is predicated of the whole may be predicated of a part (as a part of the air is air, and a part of water, water), nevertheless certain things are predicable of the whole which cannot be predicated of any of the parts; for instance, if the whole volume of water is two cubits, no part of it can be two cubits. Thus in every composite there is something which is not it itself. But, even if this could be said of whatever has a form, viz. that it has something which is not it itself, as in a white object there is something which does not belong to the essence of white; nevertheless in the form itself, there is nothing besides itself. And so, since God is absolute form, or rather absolute being, He can be in no way composite. Hilary implies this argument, when he says (De Trin. vii): "God, Who is strength, is not made up of things that are weak; nor is He Who is light, composed of things that are dim."
Reply to Objection 1. Whatever is from God imitates Him, as caused things imitate the first cause. But it is of the essence of a thing to be in some sort composite; because at least its existence differs from its essence, as will be shown hereafter, (4, 3).
Reply to Objection 2. With us composite things are better than simple things, because the perfections of created goodness cannot be found in one simple thing, but in many things. But the perfection of divine goodness is found in one simple thing (4, 1 and 6, 2).
The purpose of looking at the Attributes of God is to come into active contemplation of God. This active contemplation, obviously, brings us closer to Him as we use our reason and open ourselves to the infusion of knowledge from God, as much as He desires to give.
Sometimes, one can only contemplate God in love, when looking at the Attributes.
To be continued....