What is the significance of the Bible to Christianity?

The question here hinges on whose definition of "Christianity" is used. We could use a Reformed definition, an Arminian definition, a Roman Catholic definition, an Eastern Orthodox definition... or maybe an Anglican, or Episcopalian definition... how about the Pentecostals? The criteria for those are described fairly clearly. We could also use definitions that are less specific: how about an Islamist definition, a secular humanist definition, or a "new atheist" definition?

Edit: The difficulty with finding "the bible's definition" of who is/isn't a Christian, or of nearly anything else, is that -- even for those who believe that it has only one Author -- the bible remains is a massive book, written over the course of several centuries, in multiple languages, by many different people, living in significantly different cultures. It requires intensive, thorough interpretation. For any one definition that is offered, several more, often mutually exclusive definitions can be immediately provided -- even when all were develop by teachers that were among the most brilliant bible scholars in all of history. Pick an obscure -- or not-so-obscure -- theological concept, and look up how it was taught by, for example, Aquinas, and Calvin, and Wesley. It can get rather complicated.
/Edit

In any event, the word "Christians" refers simply to a group of people, but of those who self-identify as Christian, I anticipate that the majority would offer a more specific classification of themselves: Baptist, Methodist, Roman Catholic, Presbyterian, "Assembly of God-ite" (that's really what some AoG adherents call themselves, in a friendly, tongue-in-cheek way), etc. And each of those groups has specific criteria that can be used to determine who is, and who isn't, a member of their community. Most (all?) denominations have a formal "Statement of Faith" describing their doctrinal stance, and many churches require that any individual who wishes to join their membership first complete an extended Sunday-school type course that includes basic doctrinal instruction, perhaps over a weekend. Some also require the study of a catechism.

The centrality of the bible results from the fact that it is widely regarded as the official primary source document of Christianity.

I am aware of no groups that officially use the word "Christian," without qualifiers, to describe themselves. In practice, however, we unfortunately find that there are many who are anxious to claim exclusive ownership of the term, and denigrate any and all who disagree with their particular interpretive stance.

Ultimately, if the bible is true, it is the God of the bible who will judge whether or not any particular individual is, or is not, a "Christian."

Is Christianity defined by the Bible?

No.

If not, what is it exactly that makes a person a "Christian"? (Can an Atheist call himself a Christian and be counted as a Christian?)

Christianity is defined by belief in the Nicene Creed. Here is one version.

We believe in one God, the Father, the Almighty, maker of heaven and earth, of all that is, seen and unseen.

We believe in one Lord, Jesus Christ, the only Son of God, eternally begotten of the Father, God from God, light from light, true God from true God, begotten, not made, of one Being with the Father; through him all things were made. For us and for our salvation he came down from heaven, was incarnate of the Holy Spirit and the Virgin Mary and became truly human. For our sake he was crucified under Pontius Pilate; he suffered death and was buried. On the third day he rose again in accordance with the Scriptures; he ascended into heaven and is seated at the right hand of the Father. He will come again in glory to judge the living and the dead, and his kingdom will have no end.

We believe in the Holy Spirit, the Lord, the giver of life, who proceeds from the Father [and the Son], who with the Father and the Son is worshiped and glorified, who has spoken through the prophets. We believe in one holy catholic and apostolic Church. We acknowledge one baptism for the forgiveness of sins. We look for the resurrection of the dead, and the life of the world to come. Amen.

James T's answer is a nice analogy. The Bible guides us to find out more about Jesus Christ, about God, and to further define what these short phrases of the Nicene Creed mean.

Can you be a Christian and not believe in the Bible?

Highly unlikely, in my opinion. You can be a Christian and be ignorant of what's in the Bible.

Can you be a Christian without ever having been taught from the Bible?

I'm not sure what you mean here. You can be a Christian without the ability to quote Scripture. It's highly unlikely that you would be a effective Christian unless you've been taught by preachers or teachers that study the Bible.

If so, why is the Bible important to Christianity?

It's the guide book. The more you study the Bible, the more you learn about Jesus Christ, God, and our relationship to them.

Let me paraphrase an answer I heard from John Sentamu, the Archbishop of York, on virtually this topic, at a public lecture a few years ago. Any errors in this account are due to me, though. You asked for a Protestant perspective and I believe this counts; it is not the only Protestant perspective and perhaps some non-Protestants would also agree with it.

You're in a city at night. It's very, very dark, and the streets are deserted. In the city is a river, and across the river is a bridge. You want to get to the other side. It's possible that you might stumble across it by accident - but you could also have some guidance, like a map. The map can be relied upon but it is not the only way to get across.

Similarly, the Bible is authoritative about certain matters. What it contains is true, and it contains everything necessary for salvation. Those things are not true because they are in the Bible, but they are in the Bible because they are true. The people who set the canon recognised the truths they already knew, in written form, and put the text together in the hope that it would be useful to others. For example, you can test what other people tell you against what's written in the Bible, and reading it can lead you directly towards God.

You do not need to have read from the Bible yourself in order to have access to its truths. Other people can teach you, and so can the Holy Spirit. This worked perfectly well before the Bible was finished, and it still works now. God is also capable of leading people across the metaphorical bridge, even if they have never heard of him. But the Bible is one thing that we know "works", so we should pay attention to it, and tell other people about it.

Finally, defining who is or is not a Christian is less important than acting towards all people with love. There are certain things that we (everyone!) ought to believe and do, but it's not possible or fruitful to try to draw a sharp line between who's in and who's out (which is ultimately up to God, not us).

This may be redundant, but from a Sola Scriptura/Biblical Literalist/Protestant/Evangelical/Fundamentalist view...

• Is Christianity defined by the Bible?

  • If so, is belief in the Bible the most important (or foundational) belief in Christianity? Is it the basis of Christianity?

Yes. Christianity is the faith in the God of the Bible. If we believe in a god, any God other than the God defined in the Bible, then by definition, we are following a god that either we made up to suit ourselves, or one made up by men. Either way it's called idolatry.

• If not, what is it exactly that makes a person a "Christian"? (Can an Atheist call himself a Christian and be counted as a Christian?)

According to the rules of this site, anyone can call themselves a Christian, and we act as if that is the case. So for the purposes of this site, yes.

But: Scripture says otherwise. It teaches of false brethren, false converts - people who think they're Christian but aren't. It speaks of false teachers. It defines Christians as those who have accepted the free gift of salvation through faith in Christ alone.

• Can you be a Christian and not believe in the Bible?

Again, per the rules of this site, yes, but in reality, I don't think so. See the first portion of my answer. Further, if God is who the Bible says He is, and if the Bible is truly God's word, then to disbelieve the Bible is to disbelieve God - to think that he is either mistaken (and therefore not worthy to be called God) or a liar (also not worthy of the title.)

• Can you be a Christian without ever having been taught from the Bible?

The people saved in the New testament didn't have the Bible. It wasn't completely written yet. They were taught what would later become Scripture. So it is possible to become a Christian without having someone physically open the Book.

But: the people who were saved in the early Church were told about Christ, they were told the gospel message (as described in the book of Acts) so in essence, they were taught what the bible teaches. They were just taught it before it was recorded.

• If so, why is the Bible important to Christianity?

N/A. "If so" clearly isn't the case. The Bible is essential to Christianity today. The events described in it happened thousands of years ago, and if we didn't have this Book of history, recorded exactly the way God intended, and preserved to the present day, we'd have no solid foundation for our faith. We'd have a million different views on God (as we do today) without the one source of Truth that the Scriptures provide, as a plumb-line against which to measure the truth of the various teachings.

In other words, without the Bible, we could know nothing about God. We would have no basis for our faith, other than to base it on our own suppositions and the suppositions of those in authority who came before us.