The word ‘sacrifice’ comes from the Latin, meaning ‘to make sacred’. A sacrifice is an act of making something sacred. Typically, this means a transfer, a giving up of something to the divine.
Why give something to a god or Gods? The basic idea – as the current usage of the word sacrifice suggests – is that by giving up (or letting go of) something, there is some greater good that comes out of it or is returned.
So, when Nestor makes a sacrifice of a cow with gold-foiled horns to Athena, he believes that it will or may increase the chances of a propitious journey. This is an understanding of sacrifice within a pagan context. What of a Christian one?
A part of the understanding is still the same – by giving something up, the chance of some greater good is increased. Yet, the focus is different. Instead of a sacrifice that is intended to placate a potentially moody or capricious god, sacrifices are rather intended as a sign and instance of (‘sacrament’ of) trust in God.
For example, a common practice was to sacrifice a male, un-gelded sheep to God. Why? These sheep were intended to strengthen the flock (un-gelded), and giving one to God was a sacrament (i.e., sign and instance) of trust in (i.e., ‘faith in’) God that He would strengthen the flock. The relevant entity is not the sheep but something more like ‘the strength of the flock’, and this is not killed but entrusted to God in a symbolic and literal act of transfer or entrusting.
So, in the Christian context sacrifices are essentially about building a relationship with God through acts of trust, and in particular by mentally transferring ownership of something to God. Basically, this means accepting His guidance vis a vis whatever is ‘given up’. Sacrifice does not necessarily entail killing a, say, sheep – rather, with the case of a sheep, they were killed to mark the giving up of them to God.
The problem with the word sacrifice in current English usage is that its predominant sense is of losing something, instead of entrusting something. The latter is probably a more accurate sense of what the term is supposed to signify within Christian practice.
Sacrifices are tied to the Christian virtue of humility (you don’t control everything, and shouldn’t pretend you do) and the practice of aligning one’s will with God’s (I give this up to God to do as He wills).
In this sense, whenever we entrust something to God, we are in a way performing a sacrifice to God. Again, because sacrifice isn’t centrally about losing something, but about mentally transferring something (i.e., entrusting something) to God.
Why is the notion of sacrifice important in Christianity? One reason is that one basic idea in Christianity is that Jesus of Nazareth sacrificed himself for us, and each Mass (say) is partially a recreation (or rememberance) of this sacrifice. This sacrifice was important because it a) shows Jesus trusted God, and b) was an entrusting to God in the short term of his life and so on, in order that a greater good may come about in the longer term. It wasn’t a losing of his life, as (according to standard Christian belief) Jesus was resurrected in a glorified body after his death.
If we misunderstand what sacrifice is, we might misunderstand a basic part of Christianity. Indeed, it might seem absurd (“what was he sacrificing? he was supposedly resurrected and now reigns at the right hand of the Father!”, and so on). Central to sacrifice is not the notion of loss but of entrusting, and this is of course done because of a belief in a longer-term gain or greater good. That Jesus was (supposedly) resurrected is a sign of God’s benevolence for each of us – if we trust in Him, then in the long-term there will be a greater good not just for society in general, but also for ourselves, according to Christianity.
Again, Christianity is not about lack, but about abundance – through a trusting of (i.e., habitual sacrificing of, fundamentally, one’s ego to) God. I.e., sacrifice is fundamentally not about lack but about abundance, and therefore the symbol of the cross is not about pain and suffering but rather about what that has gained. That’s the idea, anyway.