Catholic philosopher, James Kidd, has written a fascinating and accessible article on the existence of God, published in This Rock magazine. You can read the article for yourself here: A Proof of the Existence of God. I've summarized my take on the argument below.
First of all, Kidd begins by offering a somewhat Cartesian proof of the existence of the self. Cogito, ergo sum, or "I think, therefore I am" is Descartes' oft-cited maxim. I cannot doubt my own existence without first existing.
Nevertheless, we can conceive of a state of affairs in which we do not exist. This means that in order to exist, we must possess the attribute of esse ("existence" or "being"). Another expedient example is the conception of an acorn (Kidd uses the example of a chicken egg). The difference between a real acorn and a merely imaginary one is not based on any distinction in their "acorn-ness", but simply that the former possesses esse. A real acorn exists, whereas an imaginary acorn does not.
Now, an acorn (as with any changing entity) is composed of actuality and potentiality. It is easiest to explain these terms by connotation: an acorn is an acorn in actuality, but in potentiality it is an oak tree. It is important to note that potentiality is not a thing in and of itself, but that it is a privation (an "absence" or "lack of") actuality. As Kidd points out, "a thing considered in itself contains nothing but its fullness." The nature of an oak tree, considered in itself, possesses the quality of being an oak tree, and does not possess the lack of being an oak tree.
We can now reflect on what this implies about esse (from here on, I'll refer to esse as "being"). Since every existing thing possesses being, it must be the case that being is actual. But, being cannot be composed of any potentiality, since that would require the lack of being. Hence, being itself is pure actuality.
Now we can move on to consider the attributes of pure actuality (Pure Act). Since Pure Act exists essentially, it cannot not-be - that is, it has necessary existence. Moreover, Pure Act must be distinct from everything else. The reason why is that Pure Act contains only actuality, whereas other things are composed of actuality and potentiality. Yet, if something is true of one thing and not of another, the two must be distinct. Therefore, Pure Act is distinct from other entities.
Immutability and Eternality
In order to change, a thing must first have the potentiality to change. Yet, Pure Act is not composed of any potentiality. Therefore, Pure Act is changeless. Further, whatever comes into being or goes out of being in time must have the potentiality to do so. As a result, we can soundly conclude that Pure Act exists at all times, and is therefore eternal.
If there were more than one Pure Act, then there would be distinctions between them. But, distinctions entail limitations, and limitations entail potentiality. However, it has been demonstrated that Pure Act is not composed of potentiality, so it must be one.
Everywhere that something exists, the entity in consideration must have being. As we saw above, however, every entity's existence is dependent on Pure Act. If Pure Act were not present somewhere, then nothing would exist there. Hence, Pure Act exists everywhere, e.g. it is omnipresent.
Omnipotence and Omniscience
Every entity that possesses some power and some knowledge is only partly actual. Human beings, for example, have some power and some knowledge, but we are limited in our power and knowledge. From this, we can infer that Pure Act must possess all power and all knowledge. Pure actuality is nothing less than the fullness of what potentiality may attain to.
By way of conclusion, we have seen that a single, changeless, eternal, omnipresent, omnipotent and omniscient being exists. This, as the Angelic Doctor muses, is what everyone understands to be God.