Nice question Clark. I especially liked your connection, on the first line, between the different meanings of the word sacrifice: that of ourselves, and the sacrifices we offer the gods.
It surprised me.

Similar to sacrificing the town's virgin for the gods there is a sense of a mutual sacrifice, where we are all part of a big community, where each sacrifices and we create something together. One sacrifices for one's friends, one's family, and you expect them to sacrifice, if not for you then for others.
I gave the example of the town's virgin as I think women are much more prone to self-sacrifice then men are, whether it be by staying home to take care of a sick person, or taking care of a baby. In both cases you also gain a lot, but are also willing to sacrifice a whole lot. How much you value your own life, and the value of your time plays an important role here. If you see your life as meaningless, or without purpose, you might be more likely to sacrifice.
(You give the example of Jesus. Many women have a kind of a Jesus complex where they are simply looking for a way to sacrifice their life away).

Why sacrifice? If no one would sacrifice we would all be much worse off. Society as a whole understood that if everybody sacrifices a bit then everyone benefits, or at least, as with the example above, most people. There are people who use the sacrifice of others, but don't sacrifice themselves at all. (For example, people who recruit others for suicide bombings but do everything to keep themselves alive). Those people you would want to avoid.

One should be on guard to appreciate the importance of one's own life and not sacrifice too much. For instance, if someone asks you to do something which they can very well do themselves only they prefer you waste your time instead of them theirs, it's important to notice it and, usually, say no. (See nice discussion on How do I say no?).

Your parting question, what constitutes a sacrifice? Tom Sawyer getting others to paint the fence for him pops to my mind. Did they sacrifice? They enjoyed it. That's the best kind of exchange where you can get others doing what they like instead of you doing what you hate. It doesn't always work out that way.

To take this resonance between sacrificing to the gods and sacrificing ourselves a bit further: in some mystic (Judeo-Christian) strands of thought, fasting represents a dual sacrifice of this sort because not only does it represent spiritual discipline, but it in fact involves a certain depletion of the faster's own muscle and fat, a kind of symbolic offering!

In that same tradition, the very first story of humans after the fall -- that of Cain and Abel -- underlines the importance of sacrifice, the possibility of doing it well or less well, and the potential consequences. Obviously in that context, which has deeply influenced the modern West, sacrifice is of the first importance.

Having said that, though "Why sacrifice?" and "What is sacrifice?" are enormous questions, I would say on a simple level that sacrifice has to do with a recognition that we operate in time, with a strong awareness of our own finiteness and of the fact that we can't have everything (either sequentially or at once), and making a conscious choice to value or acknowledge one option over another.  It may be something very concrete, as in: I choose to spend my time making a meal for this child, or it may be quite abstract, as in: I choose to spend half an hour on a religious ritual at the beginning of my day. But both are ways of voting very clearly for what one values and giving it value, in part, by virtue of what one gives up for it instead.

Even from this, it's clear that sacrifice can potentially be something quite impressive: a way of clearly and intentionally expressing one's view of the value of things, and a very concrete way of affirming an order of things which may be quite contrary to the accepted one. On the other hand, there is a great deal of danger associated with it as well. Asking someone to sacrifice often becomes a way of devaluting them, and in any case a sacrifice which is not chosen freely (but instead compelled by others or by circumstances) can often be fundamentally tragic, even if also somewhat heroic.