What do Linda, Charley, and Happy represent in Death of a Salesman, and what should Willy understand about each?

The characters in Death of a Salesman are complex enough to yield a number of answers to questions about what they might each represent. One way to explain what they represent is to see Linda as representing a humble and positive view of her life's accomplishments (with Willy), see Biff as a person who struggles to honestly understand his own nature and limitations (and succeeds in doing this), and see Happy as a person tortured by expectations that he cannot live up to and which drive him to bad behavior (just like Willy). 

Early in the play, a bit of dialogue expresses the way that Willy and Linda form a set of counterparts with Linda acting as a grounding force for the "mercurial nature" and "massive dreams" of Willy (as mentioned in the preface to Act I). 

Willy: You're my foundation and support, Linda.

Linda: Just try to relax dear. You make mountains out of molehills.

Later, Linda is the one who speaks to Willy's significance as a human being and also as a part of the shifting American culture of the day. She says that attention must be paid to this man, a representative in his own right of an era of striving and ambition and a brief time of plenitude that is now passing.

Linda then can be said to represent a world view that appreciates Willy's ambition and his partial successes in light of this ambition. She does not see Willy as a failure and this positive outlook would hypothetically be available to Willy too if he were to learn a lesson from his wife. 

Biff serves as Willy's antagonist, to some extent, in the sense that Biff challenges Willy to face the truth about the past and the truth about the present. Biff's own internal struggle applies to Willy as well as Biff comes to realize that he has been lying to himself all along about who he truly is. Allowing himself to believe that he was a salesman in the sporting goods store, Biff perpetuates a shared pattern of self-deception that includes Willy and Happy. He is willing to believe for a time in a pleasant lie. 

When he decides to face up to his true nature, Biff takes a step that Willy never is able to take. Biff finally sees himself honestly -- not as a failure but as a person with limitations who has made mistakes but who also has potential to achieve something in his life. He may not achieve "greatness," but he has already achieved a powerful victory over self-delusion. 

If Willy were willing to see himself honestly, he might have been able to enjoy the successes he really did have as a salesman and as a father instead of bemoaning his inability to become rich like his brother Ben. 

Happy is like Willy. He insists on placing the highest ambition as the only worthwhile goal, however unlikely or unrealistic that goal may be. The self-delusion and sense of necessity in that grand ambition connects Happy to Willy's most crippling qualities. Unable or unwilling to adjust his sense of self to actual history or to his real circumstances and try to find some modicum of happiness there, Happy (like Willy) is driven instead to define the sole possibility of happiness within an unreachable context. Yet he believes in that happiness even while he bitterly feels his shortcomings in relation to it. 

BIFF: Why don’t you come with me, Happy?

HAPPY: I’m not licked that easily. I’m staying right in this city, and I’m gonna beat this racket! [He looks at BIFF, his chin set;] The Loman Brothers!

BIFF: I know who I am, kid.

HAPPY: All right, boy. I’m gonna show you and everybody else that Willy Loman did not die in vain. He had a good dream. It’s the only dream you can have—to come out number-one man. He fought it out here, and this is where I’m gonna win it for him.

In summary, we might say about these characters that lessons of humility and a recognition of accomplishments are attached to Linda. Lessons of honest self assessment are attached to Biff. And lessons of self-delusion and the power of generic ambitions are attached to Happy.  

In Death of a Salesman, what do Linda, Charley, and Happy represent?

In "Death of a Salesman", Arthur Miller was attempting to create fully rounded characters and explore their psychological reactions to the failure of the American Dream, which Willy Loman pursued with all the unrelenting optimism of a salesperson. 

Charley is the neighbor of the Loman family and in many ways represents an American ideal. He is hardworking, successful, kind, and generous, even offering Willy a job. Unlike Willy, he raises his children to study hard in school leading to success in their careers. He is the role model of the play, especially in the way that he is clear-sighted and pragmatic, rather than a dreamer like Willy.

Linda is Willy Loman's wife and she represents the virtue of loyalty. Her understanding of Willy and their situation is grounded and reasonable. In some ways she is a perfect wife, forgiving Willy his affair, encouraging him when he becomes despondent, and keeping the household afloat. In another way, we might describe her as an enabler, feeding and sustaining Willy's eventually unsustainable life due to fear of what would happen to her without Willy. From a perspective of feminist criticism, one could argue that she is very much a victim of patriarchy, as in a society in which she could have had her own career, she would not have been placed in a position of dependence on Willy. 

Happy in some ways represents the worst of Willy's traits, following his path of blind optimism and being in thrall to the American Dream, but lacking Willy's moments of insight and unlikely to be capable of the sort of self-sacrifice we see in Willy at the end of the play. Happy's compulsive womanizing and self-aggrandisement, as demonstrated in the episode where he leaves Willy at the restaurant, shows Willy's failure as a parent and the hollowness of ideals focused on material success.