The meaning of death is limitation. We are limited beings; death is the ultimate boundary of our being. This may seem simple, on the surface, but it is actually a deeper concept. To make my point, let me offer two odd statements.
1) Death is not an event; it is an intrinsic aspect of Being.
2) Even if we are able to achieve immortality by some technological or spiritual means, our existence would still be bounded by the possibility of death.
Heidegger makes a distinction between death and demise. When a biological being stops functioning, this is what Heidegger calls the demise of the organism. Death, on the other hand, is the ever-present potentiality of Being. Even if you manage somehow to put off your biological demise indefinitely, the potential for your demise at any moment is still with you; death is still an intricate potentiality of your existence. Existence cannot be separated from non-existence. Existence stands out from Nothingness like the figure stands out from its background. You cannot have figure without ground; you can't have plus without minus, etc. You cannot have existence without death because death (non-existence) is the ground of existence.
To bring this into clearer focus, imagine for a moment that some version of the traditional notion of heaven is true. On this view, we are created by God, and God grants us immortality, which is to say, that he forever puts off the demise of our souls in heaven. (I'm just taking the notion of biological demise and applying it to the traditional concept of soul.) But even in this situation, we were thrown into existence. We did not have a choice to exist or not, we just found ourselves existing. Thus we never escape the brute fact of reality that the ground of our Being is still Non-Being. Even though we are immortal souls hanging out happily with God in heaven, the very fact that we exist implies that we did not chose to exist and could still cease to exist. As I said, this is because the very nature of existence implies the possibility of non-existence. In the case of the traditional God/heaven scenario, God is all powerful and could zap us out of existence even if, as a matter of fact, He never actually chooses to do so. Once again, the point is that death – as the potentiality for non-being – is always an intrinsic aspect of the nature of existence itself.
Personally, I don't buy the traditional notion of God. I think that "God" is just Existence, and thus God is limited in the same sense that Existence is limited, which is to say, that God did not choose to exist in the first place; God simply exists. And since God is Existence, an intrinsic aspect of God's nature is the possibility of Non-Existence. This is simply the brute logic of Being. So Death is a potentiality even for God. Even if, as a matter of fact God never actually dies, Death is still an aspect of God because the fact that God exists logically implies the possibility that God might never have existed, or might cease to exist, so even God must face the ultimate question of "Why do I exist?"
Existence is intrinsically limited by the fact that it is grounded in the possibility of non-existence. Truly understanding what this means will typically induce some degree of existential anxiety. This is good. This anxiety is a sign that you finally understand the true meaning of death.