“What is sin?” is a big question, but I think it’s a good one. What is it that makes “sin” sinful? This is especially important in light of the fact that essentially the same act can be either righteous or sinful depending on attending circumstances (a census of Israel is ordered by God in Numbers 1, but David’s census in 2 Samuel 24 // 1 Chronicles 21 was sinful; gathering manna on the Sabbath is prohibited in Exodus 16:23, but Jesus allows his disciples to pick heads of wheat on the Sabbath in Mark 2:23-28). What this means is that while there are certain acts which are always sinful (worshiping other gods, worshiping idols, committing adultery), much of what we call “sin” depends upon circumstances for its sinfulness (killing is, in fact, arguably one such act). Because of this and because of the size of the question in general, there is also danger that any answer will be overly reductionistic (i.e., will oversimplify the issue). Certainly, some modern attempts to reduce sin down to, for example, self-centeredness can be ultimately more misleading than helpful, even though there is a valid and important observation at the root of this effort (that sin is often motivated by our concern for ourselves at the expense of others).
That being said, I think you’ll find that a lot of scholars and theologians understand the essence of sin to be the violation of a relationship, the breaking of the rules (explicit or implicit) that govern how we interact with one another. So it is possible to sin against another human being just as it is to sin against God (though God takes all sins kind of personally – if I sin against my neighbor, God sees it as a sin against himself). It is even possible to sin against your own body (1 Corinthians 6:18), meaning you are violating your “contract” with your body, so to speak. You are abusing your body when your responsibility is to take care of it.
While there is certainly room for a far more detailed answer than this, to understand sin generally as the violation of the rules of a relationship is a helpful (and biblical) starting point for any discussion about sin.
For a fuller exploration of what the Bible has to say on the topic of “sin” you might also check out articles on the subject in Bible dictionaries. An older multi-volume Bible encyclopedia called The International Standard Bible Encyclopedia is available for free online. You can see its article about sin at http://www.internationalstandardbible.com/S/sin-(1).html
As to the second part of your question – “Is pride a sin” – my above talk of sin as the violation of a relationship might lead you to think that I am talking exclusively about actions rather than attitudes. But the Bible’s understanding of sin is bigger than that. It includes attitudes and thoughts. And this isn’t just with sins against God. This expansion of our understanding of sin to include the inner life and not just the outer life is a huge part of what Jesus is talking about in the Sermon on the Mount in Matthew 5-7 (especially chapter 5).
For example, if I am angry at my wife for no reason or for longer than a legitimate reason calls for, the Bible would consider this a sin (Matthew 5:22) By holding onto a slight, real or perceived, I become the one who is violating our relationship. Our relationship is supposed to be built on trust, love, and grace. When I refuse to extend grace, I am violating the trust of our relationship. Anger is not necessarily a sin, but it can be a sin and not just something that leads to sin. The Apostle Paul considers anger to be one of the hallmarks of “fleshly” thinking versus “spiritual” thinking.
Pride is another attitude that the Bible considers a sin in itself and not just something that leads to sin (though it certainly does that, too). The attitude is repugnant to God (Proverbs 16:5, among many other verses). Again, looking at sin as a violation of relationship is helpful in understanding why pride is a sin. When we are prideful, we violate our proper relationship with God and with others by exalting ourselves and esteeming ourselves as somehow more valuable than others. Often this goes hand in hand with putting others down.
It is important to note that “pride” is another big word. When we talk about being proud of our children or taking pride in one’s accomplishments, these are natural human feelings, and they are arguably good and virtuous. Pride in one’s accomplishments, however, can turn into something sinful when we lose perspective and start thinking of ourselves as more valuable than others because of those accomplishments.