Theists: What do you mean when you say that God is 'perfect'?

[Angrboda]

Maybe this is a good way of simplifying it into a quick answer:

"God is absolute in his perfection in that there is nothing lacking in his character or nature.  He does not need anything or anyone to exist. He is complete within Himself."

This is actually asserting three different things, so I'm going to respond to it similarly piecemeal.

God is absolute in his perfection in that there is nothing lacking in his character or nature.

To say that God is absolute in his perfection simply means that however God fulfills the criterion for perfection, he fulfills it in the highest degree; since this is already contained within the notion of what it means to be a perfection, that part of the phrase is just useless verbiage. On the more substantial portion of this phrase, to say that God is not lacking anything is to say that he has a full set of whatever it takes to make a perfection. This doesn't enlighten us as to what the characteristics of such a full set are, so in essence, it leaves the meaning of just what a perfection actually is unanswered. We know that to lack something is to be incomplete; that doesn't tell us what completeness consists of. More to the concerns I raised with Steve, this is suggesting that perfections are in some sense templates as to what it means to be complete. This leaves us with the disturbing question of where do these templates come from, and why these specific templates in particular. As noted, God doesn't fulfill all possible templates, only certain specific ones. And the templates themselves either come from God himself, or from outside of him. Either answer introduces difficulties for the consistency and meaningfulness of Christian theology. That these are perfections of his character or nature opens up additional cans of worms that could take several pages to unravel. (What does it mean for God to have a character or nature?) In short, your formulation raises many questions while not actually answering the original question of what does it mean for God to be perfect.

He does not need anything or anyone to exist.

The last two phrases address what is known as God's aseity, or his divine simplicity. Neo informs me that this is what at one time was considered to be the essence of being perfect, yet in some ways does not fit with our intuitions about what it means to be perfect. As I noted with Neo, the question of what God does or does not need from things outside himself is a question of the relationship between God and external things, which itself is not a part of any definition of perfection that I am familiar with. As such it seems to be a whole cloth substitution of one concept (perfection) with a completely separate and different concept (aseity), and as such doesn't so much answer the question of what a perfection actually is so much as it simply replaces our intuitive notions of perfection with a definition that is wholly other.

He is complete within Himself.

This is another, yet different, aspect of God's aseity. Much like the second entry above, it too rather misses the mark by suggesting that we simply call a concept that isn't perfection by the name of perfection, and thereby seem to have answered the question. In essence, it is simply missing the point like entry two above. Whether God's reasons are sufficient in and of themselves for God does not really address what we mean by perfection. It's an aspect of God, but not the particular one that we are interested in, presumably. In my assumption that you're referring to aseity here, this refers to the sufficiency of God's relationship to himself. As such, it also fails to address what it means to be complete, and just what such a completion (or perfection) is a completion of.

I'm interested in where you pulled this sentence from as it's a rather compact expression of a large expanse of theological questions. Is it your own invention?

Regardless, it's worth noting what Wikipedia says about aseity, if not for your benefit, for that of others reading this thread:

Aseity has two aspects, one positive and one negative: absolute independence and self-existence. In its "negative" meaning, which emerged first in the history of thought, it affirms that God is uncaused, depending on no other being for the source of His existence. In its "positive" meaning, it affirms that God is completely self-sufficient, having within Himself the sufficient reason for His own existence. The first concept derives from "the God of philosophers", while the second one derives from "the living God of Revelation" (I am who I am: Exodus 3:14).

https://en.wikipedia.org/wiki/Aseity

See also the related entry in the Stanford Encyclopedia of Philosophy on Divine Simplicity.