How do leading atheists understand morality on atheism?

Here are descriptions of morality, as understood by atheists:

The idea of political or legal obligation is clear enough… Similarly, the idea of an obligation higher than this, referred to as moral obligation, is clear enough, provided reference to some lawgiver higher…than those of the state is understood. In other words, our moral obligations can…be understood as those that are imposed by God…. But what if this higher-than-human lawgiver is no longer taken into account? Does the concept of moral obligation…still make sense? …The concept of moral obligation [is] unintelligible apart from the idea of God. The words remain but their meaning is gone. (Richard Taylor, Ethics, Faith, and Reason (Englewood Cliffs, N. J.: Prentice-Hall, 1985), p. 83-84)

The position of the modern evolutionist is that humans have an awareness of morality because such an awareness of biological worth. Morality is a biological adaptation no less than are hands and feet and teeth. Considered as a rationally justifiable set of claims about an objective something, ethics is illusory. I appreciate when someone says, ‘Love thy neighbor as thyself,’ they think they are referring above and beyond themselves. Nevertheless, such reference is truly without foundation. Morality is just an aid to survival and reproduction, . . . and any deeper meaning is illusory. (Michael Ruse, “Evolutionary Theory and Christian Ethics,” in The Darwinian Paradigm (London: Routledge, 1989), pp. 262-269).

In a universe of blind physical forces and genetic replication, some people are going to get hurt, other people are going to get lucky, and you won’t find any rhyme or reason in it, or any justice. The universe that we observe has precisely the properties we should expect if there is, at bottom, no design, no purpose, no evil and no good, nothing but blind, pitiless indifference… DNA neither knows nor cares. DNA just is. And we dance to its music. (Richard Dawkins)
http://www.simonyi.ox.ac.uk/dawkins/WorldOfDawkins-archive/Dawkins/Work/Articles/1995-05-10nomercy.shtml

The late atheist philosopher J. L. Mackie said that moral properties are “queer” given naturalism “if there are objective values, they make the existence of a god more probable than it would have been without them. Thus we have a defensible argument from morality to the existence of a god.” Agnostic Paul Draper observes, “A moral world is very probable on theism.”

If you want to learn about these issues at a deeper level, there is also a good paper by Bill Craig on the problem of rationally-grounding prescriptive morality here. My previous posts on this blog on this topic are here and here. The first one is about whether atheists can use a made-up standard to judge God for his perceived moral failures, the second one is on whether meaningful morality is rational on atheism.