How would you describe the different relationships in the novel?

One theme Austen explores in Pride and Prejudice is that of Equality of Personhood. Essentially, she is answering the question of whether connections (family or other ones) and wealth determines equality between persons in family, love and marriage, and if so, how and to what extent? She examines this through the relationships between characters, especially between:
Mr. Bennet / Mrs. Bennet
Elizabeth / Lady Catherine
Elizabeth / Darcy
Elizabeth / Charlotte
Jane / Elizabeth
Jane / Bingley sisters
Bingley / Darcy
Darcy / Wickham
Phillips / Gardners / Bennets
Bingley sisters / Gardners
Collins / Bennets
Collins / Elizabeth
Collins / Charlotte
Collins / Darcy
Collins / Lady Catherine
This list brings the unlikely central role of Mr. Collins into prominence. Collins has a relationship with each of the principals in the examination of this equality, or equilibrium and disequilibrium, theme. Collins has a significant relationship with Mr. and Mrs. Bennet, Elizabeth, Darcy, Charlotte and Lady Catherine de Bourgh. To illustrate his central, or hub, role, Elizabeth has a relationship with Mr. and Mrs. Bennet, Darcy, Charlotte, Lady Catherine and Collins: Elizabeth, the primary protagonist, has a circle of relationships that shares key points with the circle that Collins has. This leads to the revelation that Collins is intended for more than a caricature of ridiculous comic relief. He is intended as a dynamic character who has a central role in the examination of connections and wealth touching upon the question of equality of personhood in relationships of love, family and marriage.
Elizabeth and Lady Catherine
[Lady Catherine,] "If you were sensible of your own good, you would not wish to quit the sphere in which you have been brought up."
[Elizabeth,] "In marrying your nephew, I should not consider myself as quitting that sphere. He is a gentleman; I am a gentleman's daughter; so far we are equal."
"True. You are a gentleman's daughter. But who was your mother? Who are your uncles and aunts? (Ch 56)
The relationship between Elizabeth and Lady Catherine is perhaps the one that most easily illustrates the thematic question of equality of personhood in relationships. As the quotation above shows, Lady Catherine vehemently questions whether Elizabeth could ever attain equal personhood with Mr. Darcy because Elizabeth has an inferior mother and, through her, connections to inferior "uncles and aunts":
"But who was your mother? Who are your uncles and aunts?"
Lady Catherine grants that Elizabeth's position in the upper class as "a gentleman's daughter" is undisputed, yet she strongly questions that that position alone is enough to grant Elizabeth equality of personhood with Darcy. Lady Catherine implies strongly that Elizabeth's family connections are enough to undermine Elizabeth's personhood and create disequilibrium between her and Darcy, and certainly between Elizabeth and Lady Catherine, who, through Elizabeth's marriage to Darcy, would then be connected--much against her will--to Elizabeth and, through Elizabeth, to the Philips of Meryton and the Gardners of Cheapside, London.
To refresh readers' memory, Mr. Bennet is in the upper class as a landed gentleman of independent wealth (though now that wealth is all but squandered), but he married a woman in a lower class from the Gardner family. He married beneath his social class. This worked to Miss Gardner's benefit, now Mrs. Bennet, because her marriage to him raised her and her children to the upper class. Nonetheless, her relations--Mrs. Phillips and Mr. Gardner--remain in a lower class. This is the connection which Lady Catherine queries Elizabeth about and to which she so strongly objects.
Austen leads us to her understanding of a true perspective on the question of equality in personhood through Elizabeth's reaction to Lady Catherine at the instance of their first meeting. Visiting at Hundsford parsonage, Elizabeth is anxiously briefed by Mr. Collins on what to expect from Lady Catherine and on how to properly deport herself before Lady Catherine. With this building suspense, Elizabeth finally climbs the steps of Rosings and encounters Lady Catherine and Miss de Bourgh, accompanied by her governess ("the three ladies before her"). The narrator reports on Elizabeth's reaction to the initial encounter:
Elizabeth found herself quite equal to the scene, and could observe the three ladies before her composedly. (Ch 29)
Elizabeth is "equal" to the encounter, including metaphorically and symbolically equal. In terms of this theme, Elizabeth finds herself equal in personhood to a titled lady if not equal in wealth or connection. Austen is suggesting that within equal classes connection and wealth do not affect equilibrium or disequilibrium of personhood in relationships. The relationship between Mr. Collins and Lady Catherine presents the converse of this perspective.
Mr. Collins and Lady Catherine
Mr. Collins as a clergyman (clergy are in lower classes) believes that the office of the clergy elevates him to the level of the highest persons in the land, as he tells Elizabeth at Bingley's Netherfield ball:
"give me leave to observe that I consider the clerical office as equal in point of dignity with the highest rank in the kingdom" (Ch 18).
Yet, if it weren't for Lady Catherine bestowing an important "living" upon him (living: the pastoral position and income granted by a benefactor to a clergyman in Great Britain), he would not have the connections or income to back up that claim: his parish might as likely be poor and insignificant as well-to-do. Austen uses Collins' unique position in relation to external connections to illustrate the theme of equality of personhood even more clearly.
Charlotte, the daughter of a knight, Sir William Lucas, marries beneath her upper class status--just as Mr. Bennet did--yet, because of Collins' connections to Lady Catherine, Charlotte's material position is not lessened but actually raised. She has access, though limited, to the highest level of upper class society. She has a life of sufficient ease and comfort. She has gained independence from her position as a dependent unmarried daughter. She has gained social prominence, influence and importance in her new parish. Collins' connection to Lady Catherine has (1) created greater equality of personhood for him, thus raising him to Charlotte's level, enough for her to consider a marriage to him, while simultaneously (2) having elevated Charlotte's own position in society to one of greater equality of personhood by bestowing on her importance, connections and influence.
This equality is a result relating to the question of equality of personhood that Elizabeth could not have predicted. Indeed, it is one that, even as a visitor, she cannot see even though Charlotte does [the narrator's report of Elizabeth's ironic thoughts upon departing from Charlotte provide one instance in which Elizabeth is presented as an unreliable character: her opinion of Charlotte's choices, "Her home and her housekeeping, her parish and her poultry, ... had not yet lost their charms," may have been entirely different if solicited after having learned to shed her prejudice]. The many other relationships between characters further examines equality of personhood in all its varying shades, degrees and kinds.