What is God's nature?

In His own words:

Exodus 3:14 (NIV)

God said to Moses, "I AM WHO I AM. This is what you are to say to the Israelites: 'I AM has sent me to you.'"

And in centuries of sudy, we still only have a surface understanding of what that means, who He is, and his essence. We have, from Scripture, everything we need to understand what He wants us to know about Him, but even in a lifetime of study, our understanding changes somewhat with each pass through the Bible.

The simplest terms? There can be only one answer, I think:

God is love

1 John 4:8 and 1 John 4:16b.

This is the very heart of who the Christian God is. To answer the second part of your question; it defines how He interacts within the Trinity¹:

... a voice from heaven said, “This is my Son, whom I love; with him I am well pleased.”

Matthew 3:17

and with His creation:

For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life.

John 3:16

¹_{It also shows why there has to be a Trinity. God cannot be love if He is just one Person. Instead, it requires one Person to love Another, and that Love between them is Another (source and also Romans 5:5).}

This, being not a philosophy site where all possible natures are to be debated, I will endeavor to simply state the most fundamental ways in which Christianity knows to describe God. This is, at best, a nearly futile exercise, but we have been given a few guides. First as an illustration of how words will never do him justice, note how he chooses to introduce himself:

Exodus 3:14 (ESV)
¹⁴  [...] “I am who I am.” [...]

No explanation. No description. No more. Just "I AM".

However, he doesn't leave us hanging. Most significantly, the "I AM" revealed himself in both time and space in the person of Jesus Christ, "The image of the invisible God". Through him we catch a glimpse of God, and through his own teachings and affirmations of the OT scriptures we arrive at various ways of expressing what sort of thing God is.

John 4:24 (ESV)
²⁴  God is spirit, [...]

Likewise we know much about his works, that in turn tell us more about what he is. For example, we know he is the creator of all things. This is referenced many times in Scripture, but here is the corresponding statement from the Nicene Creed, one of the basic statements about what Christians believe.

I believe in God, the Father Almighty,
the Maker of heaven and earth

Likewise we also know some things about his Character which help fill out the picture. For example we know that he is able to have personal relationships and what characteristics he shows in those relationships.

1 John 4:16 (ESV)
¹⁶  So we have come to know and to believe the love that God has for us. God is love [...]

Yet "love" is just one of his attributes. It is hard to say what is most fundamental about him, but if anything, the only attribute that is constantly emphasized by being repeated three times, would be his holiness.

Isaiah 6:3 (ESV)
³  And one called to another and said: “Holy, holy, holy is the Lord of hosts; the whole earth is full of his glory!”

From this (and many other Scriptures) we get the idea of a perfect God who transcends many of our measurable attributes like time or space, yet who interacts with physical creation and carries on a relationship with those he has made.

A more detailed statement comprised of information from many parts of Scripture can be found in the many Creeds and Confessions, such as this except from the Westminster Confession:

There is but one only living and true God, who is infinite in being and perfection, a most pure spirit, invisible, without body, parts, or passions, immutable, immense, eternal, incomprehensible, almighty, most wise, most holy, most free, most absolute, working all things according to the counsel of his own immutable and most righteous will, for his own glory, most loving, gracious, merciful, long-suffering, abundant in goodness and truth, forgiving iniquity, transgression, and sin; the rewarder of them that diligently seek him; and withal most just and terrible in his judgments; hating all sin; and who will by no means clear the guilty. WWCF, Chapter 2, Point 2

It would be remiss to touch on this subject without also mentioning the Trinity. God as he has revealed himself to us is but one being yet he has three persons by which we know him. Again we arrive at this picture through a multitude of references and the words of Jesus himself, yet it is very hard to put into words. One of the most valiant attempts I know of is The Athanasian Creed, excerpted:

That we worship one God in Trinity, and Trinity in Unity; Neither confounding the persons, nor dividing the substance. For there is one Person of the Father, another of the Son and another of the Holy Spirit. [...] So the Father is God, the Son is God, and the Holy Spirit is God; And yet they are not three Gods, but one God.