There are many divine attributes (aspects of God's character, or descriptions of what God is). For example, these "classic" attributes:
Theologians have always taken pock shots at these divine attributes, even in the earliest days of the Church. (For instance, in what way can God be unchanging in Exodus 32:14?) In their own way, so do regular folks, just by the way they live their lives and face life's tough situations. The acts of daily living raise questions about the very character of God. For instance:
God doesn't get upset at these questions, and it's not a sin to ask them - even in angergrrrr.... They get at the heart of the mysteries of faith trust, justice, daily life, and even of existence itself. To speak of the attributes of God is to say something about God's nature, and the only things we know about it are what God has shown and told us in Scripture, and shown to us through Jesus. We have the information that really matters, more than enough to live by, yet we're still dealing with something beyond our abilities to describe or understand -- which means that, when pushed far enough, even the most truthful of the words, descriptions, images, and theologies we use to describe God will miss the mark bad aim. These questions defy simple answers. To people who look deeper, the attributes of God point us to how the divine attribute definitions miss, so we can get closer to the core of what lies behind them -- the awesome character of God.
The honest truth is, though, that our questions about the character and attributes of God also reflect in our own questions about humanity. For instance, every question about a gracious God who allows war is more of a question about the leaders who give the orders to fight wars, the war apologists who propagandize war, the generals who strategize war, and the soldier that pulls the trigger or throws the grenade -- all of whom are humans with the freedom to decide whether to do it or not. This effect on us makes the attributes too important to ignore.
"Among the attributes of God, although they are all equal, mercy shines with even more brilliancy than justice."
appropriation : a Trinitarian teaching which states that while all three persons of the Trinity are active in all Godly doings, it's okay to think of each such action as being primarily the work of one of the Three. For instance, creation can be seen as the Father's work, or redemption cash in as the Son's work, or enlightenment as the Spirit's work workoholic God, so long as it is understood that all three persons are present and at work in all of these things in some way, working toward the same purpose.
A related concept is that of divine unity. Put simply, all persons of the Trinity are equally God with the same divine purpose, and have all of the attributes of God (see below) equally. For example, we think of Jesus as merciful, but the Holy Spirit and the Father are as merciful as Jesus. We think of the Father as transcending time, but so do the Spirit Holy Ghost and the Son.
hypostatic union: [Greek hypo- (under, beneath) + stasis (standing, positioning, state)].
In the Athanasian creed, roughly two-thirds of it is taken up with a description of the underlying or foundational (hypostatic) unity of the Trinity, mostly done by saying things and partly taking them back. What holds it together is more than mere hypostatic cling - indeed, what holds God together holds all things together. That three are one is by definition a 'relationship group hugs?' -- yet being one in substance, one in purpose, one in love for God's creatures, implies that something more marks them as a single entity.
Almost anything we say about God's internal makeup is bound to cross over the line into falsehood. (That's why the church of the 300-500 AD period was so keen on questions of how Jesus was human but also God no one's right. One wrong turn, and the church would have lost its way.) Such untruths usually tread closely to some part of the truth. We're using words to describe what makes God God -- but isn't God too far beyond us for that? Yes and no. God put into His followers the drive to know and understand God as much as they can, even with their feeble capacities for it. That's what we do over someone we love - we hunger to know the loved one better. And God revealed what was most important, most central, most definitive, most characteristic about God, firstly in the Bible the source, then definitively in Jesus. Know Jesus, and you'll know all you need to know about God's inner workings.