Oliver's mother, Agnes Fleming, does not actually appear for more than a couple of scenes at the very beginning of this novel as she is giving birth, but the story of how she came to be pregnant, unmarried, and giving birth in a workhouse ultimately turns out to be very important for Oliver. The surgeon here assumes he is seeing the sad outcome of "the old story"—a woman being seduced outside of marriage and then abandoned by her lover once pregnant, as there is no wedding ring. An unmarried pregnant woman in Victorian England would have been deemed a social outcast, and as such it was sadly natural that a workhouse surgeon would have seen many such women reduced to dire poverty.

Later in this novel, we learn that the surgeon was, for the most part, correct in his assumptions. Agnes, Oliver's mother and the sister of his future caregiver Rose Maylie, was seduced by an older man, Leeford. However, although Leeford doesn't marry his mistress after impregnating her, he does leave a will stipulating that all his worldly possessions should be left to Agnes and her offspring, so he did not intend to leave them completely unprovided for. Unfortunately, Oliver's mother dies before he can learn who she was or who his father was, and because the surgeon assumed that this was simply a case like any other of "the old story," no attempt is made to discover the identity of Oliver's father. Not until he has suffered a childhood of deprivation, plus attempts by his half brother "Monks" Leeford to steal his legal inheritance (of which Oliver himself knows nothing), does Oliver succeed in escaping the grim circumstances of his early life.

This quote refers to Oliver's poor mother and the way in which she met her death whilst giving birth to her son, the aforementioned Oliver Twist. The quote you have identified concerns the somewhat ignoble situation of Oliver's mother. She, suffering from reduced circumstances, ended up in a work house, which was a kind of Victorian charity house for beggars who could not look after themselves. She gives birth in this workhouse and as such receives minimal medical care from the surgeon. However, because of the nature of the workhouse, many women go there who have become pregnant outside of marriage. This is what is referred to as "the old story" by the surgeon as he tries to see what identifying marks there may be on the corpse of Oliver's mother. The fact is that to have a child outside of wedlock was such a social stigma that many women fled their homes and families to give birth in secret. The surgeon assumes that this is another such case.