Defining morality is easy if you approach the task concurrently from three angles: from its appearance, from its functionality, and from the value we attribute to it.
Morality comprises first of all sacrifice. Admirable sacrifice.
Morality comprises most amount of selfless good minimizing selfless bad. Good and bad here adhere to how the moral act affects the social or if you like, wider, biological environment.
Morality is sourced from an inner urge, it is not possible to resist moral behaviour.
Morality -- when you really think about it -- always commands one to, ultimately, preserve the possibility of the self's DNA or the DNA of the closest approximation to that of the self, to survive, prosper and propagate. --------------
Humans and cats save their children from fire. We actually walk into fire to save our children. Are we, cats and humans, wont of regularly walking into fire? No, not the least bit. But walking into fire to save our offspring gives us a better chance of having our genes propagated down to posterity than not walking into fire. So we do walk into it.
Generally, a moral person will always sacrifice some benefits due to the person, in order to maximize survival chances of those who can carry on his or her legacy of DNA.
A human will kill a dog to save a human, a human will remove a drowning dog from water, a human will sacrifice his own child for god.
This part has spoken for the practicality of morality.
How does it feel though? We, humans, like all other species, have behaviour compelled by our urges. For instance, we eat because we feel hungry, not because we consciously figure out that food sustains us. (Exceptions exist, like always: I used to have a bulimic girl friend, who had go so unused from normal eating, that she had to figure out her daily caloric and other nutrient intake on charts, (at first, later from memory) and eat the calculated amounts and types of food to sustain herself. She had no sense of biological feedback mechanisms to guide her self-sustenance. She had a good figure, too.) Similarly, our moral behaviour patterns and acts are not driven by what philosophers tell us, but inner urges tell us.
In this aspect, inner urges will tell us to share our bread, to save the drowning, to save our children from fire. We do these things without thinking. We are not even compelled to do these acts, they are automatic. (A compelling feeling has a resistance to overcome; there is no resistance to our own actions when we act morally.)
A moral action is viewed as a noble act. This is a group phenomenon. Moral behaviour is promoted by groups, because in evolved nature there is always more than one feedback loop to guide behaviour (be the behaviour voluntary, or involuntary). Voluntary-seeming moral behaviour is guided by our inner urges, and is reinforced by a secondary, group-based feedback loop. We even give awards to moral behaviour, sometimes posthumously, to affect those who survived the lifespan of the saints. A third feedback loop would be also group-related, but involves the help, invokes the help, of supernatural beings -- gods, to be exact. Pleasing a god is good, moral behaviour.
My own, private, definition of moral behaviour rests on the thesis that all humans 1. crave eternal life and 2. are willing to do anything to attain it. One way to eternal life, so the belief goes, is always acting morally. Another way, to act selflessly. These unfortunately oppose the survival tactics we need to employ in life. So life is a struggle of keeping a balance between survival-benefitting selfish behaviour, and survival-detrimental unselfish behaviour. Some morality is socially heavy, some is individually heavy. For instance, rape and pillaging is socially unacceptable, but feels good to do. Sharing your bread, working for the common good, fighting for your fellow human's rights feels good, but not in a selfish way. After pillaging, or eating a lot at someone else's expense, you go home with the feeling that you "cheated the system". After a day of community street cleaning, or church fundraising, or charity work, you go home with a good feeling of having paid your dues. You can't survive with just one or the other type of behaviour.
In my private opinion, an inner moral guide will tell each human how to behave morally. There is no resistance to moral behaviour. If you want to describe or refer to the core of moral behaviour, I say it's a behaviour that the individual feels will help him or her get to a pleasant afterlife. Selfish behaviour, expecially on the expense of denying or violating moral behaviour, tends to tell people that they will go to hell. (So to speak.)
Heaven and Hell are therefore not exclusively Christian-created concepts. In pre-christian religions we find no references to them; but behavioural references always existed between good and bad morals, and those included the need of one to serve his or her god.
Christianity's developing the concepts is not an innovation, but an innovative use of human's inner moral world views. Humans always think and have and had thought of things as "good" and "bad"; Christianity simply institutionalized these notions into the conceptual creation of Heaven and Hell.
I talked enough.