It is beyond the scope of an eNote answer to explicate an essay of this length line by line, but I will provide an overview, and please feel free to ask more questions.
As the title of the essay indicates, the subject is poor relations, by which Charles Lamb means relatives with very little money. They were common in early 19th century England because society favored the accumulation of wealth into a few hands. For example, the laws of primogeniture ensured that great estates were inherited in their entirety by the eldest son, rather than divided among the children. This kept the estates intact and maintained the family prestige. Custom, as Samuel Richardson had outlined a half century earlier in Sir Charles Grandison, also tended to favor leaving even discretionary income to one heir, something Richardson deplored as cruel to other relatives. Jane Austen's novels, close to Lamb's period, also dealt with the issue: the plots of both Sense and Sensibility and Pride and Prejudice are put into motion by the real or threatened loss of an estate to a male heir.
What kept wealth concentrated in a few hands (the good fortune of eldest sons) tended to leave other relatives in more precarious situations that could lead to poverty, especially given the lack of a social safety net in that period. Then, as now, having the poor relation show up for dinner could be embarrassing (if you have seen the movie National Lampoon Vacation, you will remember the comic problems that arise when poor, uneducated relations show up), and this becomes the focus of the Lamb essay, though more poignantly than in National Lampoon: the poor relative who arrives at the house for a meal.
In the first paragraph, Lamb, in the guise of his narrator, Elia, lists some of the common thoughts or cliches about a poor relation, none of them flattering. A poor relation is "irrelevant" or unimportant, "impertinent" in writing to you, in other words, a person you don't want to hear from, a drain on your finances, an embarrassment, something that delights your enemies, inconvenient, annoying, a blot on your life. These are blunt words, and while there is a comic intent in this piling up of cliche upon cliche to the point of exaggeration, at the same time, Lamb/Elia doesn't fall into silly pieties or the hypocrisy of pretending a poor relation is wholly welcome.
In the second paragraph, Elia describes the arrival of the male poor relation, including the mutual embarrassment: you, the host, really don't want to see him and he is embarrassed by his poverty, but he needs a free meal. (We remember that the problem of food scarcity isn't really solved, even in rich countries, until after World War II, well beyond this time period.) The poor relation never seems to arrive on the days when you don't have other company. He, of course, shows up at dinnertime, then has to be persuaded to eat the food he is hungry for, even if there is none too much ("the turbot ... small"). Lamb lays out the whole embarrassing scene:
He declareth against fish, the turbot being small — yet suffereth himself to be importuned [persuaded] into [taking] a slice against his first resolution. He sticketh by the port — yet will be prevailed upon to empty the remainder glass of claret, if a stranger press it upon him.
His manners (how he acts) are also excruciating: he is both too "familiar," in other words, he acts too much like a close friend, and at the same time, he is too "diffident" or shy, too abject and humble. The servants don't know how to treat him and the other guests wonder about him, though his unfortunate knack of being both overly friendly and overly abject betray him as the poor relation. He brings up old family stories at the wrong moment (is "unseasonable") and his conversation and compliments irritate (they are a "trouble" and "perverse"): his presence, in a word, is awkward, and when he's gone, you whisk his chair into a corner and breathe a sigh of relief. He's that person who doesn't fit in but who you can't not have over.
Elia then moves to the female poor relation. He finds poor females worse than poor males because they have an even more difficult time hiding their status and can't be dismissed easily as simply eccentric:
But in the indications of female poverty there can be no disguise. No woman dresses below herself from caprice. The truth must out without shuffling.
Her clothes are between those of gentlewoman and a beggar, presumably meaning once well made, stylish clothes of good fabric now worn, outdated and patched. But even worse are her manners. She is too humble, too self-aware of being a poor relation, too abject, and people hold her in contempt: for instance, the governess, below her rank, corrects her when she calls the piano a harpsichord (a humbler, more old-fashioned instrument).
In the next paragraph, Lamb first mentions a Richard Amlet, a poor gamester (or gambler) in a play called "The Confederacy" by Sir John Vanburgh, then moves to a friend, who was the son of a house painter. This man went to Oxford and loved the scholarly life, but was pulled out of it to take up his father's trade, for financial reasons. The difference between "gown and town" or academics and life in a shop, was too great for this young man, who instead joined the army and was killed in Portugal.
Elia goes on to say in the next paragraph that while he started off his essay half comically, the subject of poor relations is also painful and tragic. He mentions his own childhood and a poor relation, an old gentleman, who would come to dinner every Saturday and one day was offended when Elia's aunt pushed a second helping of food on him saying, “Do take another slice, Mr. Billet, for you do not get pudding every day." He gets revenge later on her calling out his poverty by labeling her superannuated, which means obsolete or outdated. But Elia allows the poor relation to land on a note of dignity, for this elderly man dies poor, but with five pounds to his name: "enough to bury him." The very ending, however, is ambiguous, possibly ironic: he had "never been obliged to any man for a six-pence" (if you don't count the weekly meals).
Overall, the essay is notable for dealing honestly with an uncomfortable subject.