Although he's not commonly known today (and was unheard of in the Silver Age), the villain in these two issues is very significant in Batman's history, as he's the prototype for much of what would follow.
1. Is the first villain to set a death trap for Batman. The death traps became something of a cliche, especially during the Batman TV show.
2. Is the first recurring villain.
3. Is the first villain to have a huge henchman, something that became common in the years to come.
4. Is the first villain to become horribly disfigured, something that would happen to major Batman antagonists like the Joker and Two-Face.
5. Is the first villain to recognize that he has to plan on Batman interfering with his operations.
Note that despite all those first, Dr Death is not the original villain with a monocle and Van Dyke beard; that honor went to Frenchy Blake in Tec #28. He is troubled by the possibility of Batman interfering with his schemes, so he puts a personal ad in the paper, telling Batman to check at the post office for a letter addressed to John Jones and:
This illustrates another difference between the early Batman stories and what would come later. Because Batman was wanted by the police, he could not simply go around investigating on his own, and Bruce often did some of the legwork. Indeed, this was the whole rationale for having Bruce be a friend of Commissioner Gordon.
The letter tells Batman that the writer plans to commit a murder on the 14th floor of an apartment building. Batman climbs up the outside of the building using suction cups on his hands and knees (an invention that would not appear again as far as I'm aware). When he gets to the top, he discovers that he is to be the murder victim:
As death traps go, it wasn't very elaborate, but every tradition has to start somewhere. Jabah shoots Batman and in an amusing scene, Bruce gets treatment for his wound from the family physician. Later, he spots Jabah about to kill a man with a deadly powder:
This is Dr Death's plot. He threatens to kill wealthy men with the powder, and if they do not pay his protection money, he sends Jabah out to slay them. Bruce saves the man's live by placing a handkerchief over his mouth, and trails Jabah back to Dr Death's hideout, where he changes into the Batman. He disposes of the henchman with a rope around the neck (possibly killing Jabah). Then he and Dr Death have their battle, during the course of which, the laboratory catches fire and:
Dr. Death apparently dies in the fire. However, he returns in the next issue, making him the first "resurrected" villain, although far from the last. Comic book writers often kill of the villain at the end of a story as it saves time in the denouement and also makes it ironically easier to have the villain return since there is no need for a parole, just an explanation of how he escaped death.
In the following issue, Bruce's attention is drawn to the story of a man who died suddenly. Sure enough, when he confers with the widow:
Bruce returns as Batman that night to temporarily steal the widow's diamond collection (for safekeeping). But Dr Death's backup henchman, Mikhail, is also after the diamonds, and Batman lets him take them so he can follow him back to his boss. After dropping the jewelry off at a fence, Mikhail returns to a flophouse. They have a fight, during the course of which, Batman escapes through the window to his waiting rope. And:
I'd take that as pretty strong evidence that Batman killed both Jabah and Mikhail. He goes back to the fence's residence, where he deduces the man is actually:
Dr Death would return many years later (1982) in a two-part series patterned on the prior adventure with him infecting millions of Gothamites and holding the city for a one-billion dollar ransom.