Milverton is a famous blackmailer that Holmes finds particularly despicable. This is interesting since Holmes deals with murderers, thieves, con-artists, etc. What makes Milverton so awful? We'll let Holmes explain it:
"I would ask you how could one compare the ruffian, who in hot blood bludgeons his mate, with this man who methodically and at his leisure tortures the soul and wrings the nerves in order to add to his already swollen money-bags?" (Milverton.10)
It all comes down to intent here. Milverton plans his crimes and delights in tormenting people for money. Holmes finds this sort of cruel intention much worse than someone who commits a crime in a moment of passion. But Holmes also finds Milverton an intriguing opponent. Milverton frustrates Holmes, but he also challenges him. Milverton is sort of like an anti-Holmes, a super smart man who uses his powers for evil, not good.
In the end, the Milverton problem drives Holmes to actually commit a crime himself, by breaking into Milverton's house, which is extreme even for Holmes. Holmes and Watson end up witnessing Milverton's murder, which isn't all that surprising given how many people the man blackmailed and how many lives he ruined. What is interesting though is that Holmes himself never really "beats" Milverton. This definitely stands out among stories where Holmes usually saves the day and wraps everything up nicely.
This is the client who gets Holmes involved with Milverton. Milverton is blackmailing her with some love letters she sent years before; he's threatening to show the letters to her wealthy fiancé. Holmes's desire to help Lady Eva drives him to extreme measures when dealing with Milverton. Well, partially at least. Holmes's hurt pride probably had more to do with his desire to take down Milverton. Holmes is definitely a sore loser.
This is the man that Lady Eva is set to marry, if she can avoid being blackmailed that is. The Earl is from an old, wealthy family that frowns on the slightest hint of scandal, which is why Lady Eva is so scared about him discovering the blackmail.
Agatha is a maid in Milverton's house that Holmes, for lack of a better term, seduces. Holmes gets fake engaged to Agatha and he pumps her for information about Milverton's house, which later help Watson and Holmes during their home invasion. We never hear about Agatha after this, and Holmes seems to think that the fake engagement is no big deal since she has another suitor who will likely actually marry her.
The Milverton story is one of the most unique Sherlock Holmes stories, and this character is yet another reason it stands out. We never learn the name of the woman who murders Milverton in revenge. This is deliberate too; Watson refuses to tell us since he's protecting this woman. Both Watson and Holmes keep her identity a secret, since they can't reveal her without confessing that they broke into Milverton's home and witnessed a murder. But the two men also seem to think that this woman was justified in what she did. As she told Milverton before killing him, she was seeking revenge on her husband's behalf, since Milverton's blackmail of her drove her husband to despair and later death. This character is also a prime example of the sort of flowery, over-the-top language that Watson often uses to describe women.