him to be an experienced and intellectual man. He was an admired practitioner of faith. Yet among these great qualities, he chose a timid and young spiritual mentor. This is odd. He attached himself to this young man and must have a reason for it.

About this period, however, the health of Mr. Dimmsdale had evidently begun to fail.

Conveniently, the young reverend had very recently made the acquaintance of a leech. The ill in him is doing him more harm than good, just like the effort of leeches in those days to indeed suck out the 'bad blood.'

The actual leech is a worm from the annelid family which has suckers at both ends of its body. As such, this invertebrate attaches itself to the skin of vertebrates, sucking the immediate blood source that comes up in the process. Leeches produce a natural anticoagulant and blood thinner. This is the reason why leeches were the preferred method for "curing" people many centuries ago. However, leeches are making their way back into modern medicine.

Back to the story, in the time period where The Scarlet Letter is set (17th century) a physician such as Chillingworth (who pretended to be one) would have been referred to as a "leech" precisely because of the employment of the worms during treatments. However, this is also a figurative term that applies quite well to Chillngworth, who seems to be sucking away at the soul of Dimmesdale with his constant questioning and mentioning about what could be lurking within the soul of the priest. At first, the people saw Chillingworth as a blessing for their beloved and sickly Reverend Dimmesdale. 

This learned stranger was exemplary, as regarded, at least, the outward forms of a religious life, and, early after his arrival, had chosen for his spiritual guide the Reverend Mr. Dimmsdale.

However, after a while, the true intentions of Chillingworth, his inner anger, and his hunger for revenge began to reflect in his face, making him seem evil and scary to all the villagers. We know as readers that Chillingworth, upon discovering the situation that Hester was under as a pariah of the village, was determined to bring out who it was that wronged him. Chillingworth is willing to do anything for Hester and her lover to come forward and tell the truth that it is Dimmesdale who fathered Pearl. Hence, to disguise himself as a "leech" and, literally, latch on to the body and soul of Dimmesdale would be the easiest way to make his plans work. 

A leech is a blood-sucking parasite, and it is also used to describe a person who sponges off or uses other people.  Therefore, it is a perfect word to describe Chillingworth because he is figuratively sucking the life out of his "host" (so to speak), Arthur Dimmesdale. Dimmesdale is getting weaker and weaker, as though his life is ebbing away. Chillingworth believes, correctly, that Dimmesdale is Hester's co-sinner and the father of little Pearl, and he has vowed to ruin Dimmesdale and make him pay for the harm that the minister has done to him (by sleeping with his wife). Chillingworth is sort of living off Dimmesdale, his hatred and need for revenge keeping him alive, as, when Dimmesdale dies, Chillingworth dies shortly thereafter (just as a parasite would not be able to survive without its host).

A leech is a blood sucking insect.  Very unattractive little guys.

A leech is also a homophone for a person who attaches themselves to you, like Chillingsworth does to Dimmesdale.

An actual leech is a semi-parasitic animal. There are no leeches in The Scarlet Letter, so the chapter title "The Leech" serves as a metaphor.

Real leeches have a dubious reputation. They are jawless relatives to earthworms, and some are slimy and found in swamps. Most memorably, they are capable of latching onto a larger animal such as a human and sucking blood out of it. However, they were historically used for medicinal purposes as well, owing to ancient Greek misconceptions about the role of blood in the human body. The word "leech" is derived from a Latin word meaning "physician", but owing to their bloodsucking ability, the word has become a synonym for a person who behaves parasitically, i.e. profits through harming others.

"The Leech", as a chapter title, is referring to such a person; Roger Chillingworth. Ostensibly this is because of the original Latin meaning of the word, i.e. doctor, because Chillingworth has disguised himself as one. However the double meaning is revealed through Chillingworth's inhabitance with and "care" for Dinnesdale, implying that Chillingworth has attached himself to Dinnesdale and is metaphorically sucking the life out of him;

[Dinnesdale] was haunted either by Satan himself, or Satan’s emissary, in the guise of old Roger Chillingworth. This diabolical agent had the Divine permission, for a season, to burrow into the clergyman’s intimacy, and plot against his soul.