How does Everyman describe the nature and destiny of humankind?
By nature, humankind wants to avoid death's call, and everyman (we) spend most of our energy absorbed in activities that help us avoid thinking about what happens when we die. However, according to Everyman, there is no escaping death, and there is no escaping being called into account for what we've done while we've been alive. When death comes, Everyman is unprepared: he thinks he has more time, and tells Death he is not ready to accompany him: none of us are. Certainly, none of Everyman's earthly companions are either. Each, in turn, agrees to be with Everyman until he finds out where Everyman has to go; then each, in turn, refuses to accompany him. Just like Everyman, humankind must all face Death alone. There is no avoiding the final call, either, and no postponing it until a more convenient time. When Death comes, everyman even tries to offer him money to come back later, but no earthly goods have any effect on Death. It is our destiny to die, and we will all die when God sends Death to collect us. We should, therefore, be prepared, for the only thing we can bring with us when we meet our reckoner is the accumulation of good works that we have accrued in our lifetimes (according to Everyman).
Everyman describes the nature and destiny of mankind both in a general fashion and in a specifically Christian fashion. Speaking generally, every human being will die, and almost all human beings face a day of reckoning of some sort. Even the completely secular and atheistic have moments when they reflect on their lives and seek meaning. However, in a Christian sense, the meaning is more pointed. Everyman shows us that we will all die and face a specifically spiritual judgment. When we do, the many things that we value in life, like Goods and Strength, will prove wanting. We'll need to turn to spiritual help at that moment if we want to go to heaven, rather than burning forever.
In addition to this answer, Everyman is sent on this journey for making poor choices. God is angry with man's decisions, and sends him toward his judgement. It is the nature of mankind to deny wrong doing, and then to try and beg more time--both of which Everyman does in the play. Then, once he accepts his journey, he attempts to get others to accompany him. Also true to human nature, all his "friends" agree to go until they find out where he's going...suddenly everyone has to wash his hair or walk the cat.