Willoughby is the young man who is mysteriously murdered at the start of the story. Willoughby was an assistant to an old Professor who lived in a very secluded household. This poor guy was really in the wrong place at the wrong time; his death was a result of panic, not premeditation. While many of the murder victims in these stories either had it coming, like Peter Carey, or were involved in something shady themselves, like Pietro, Willoughby was a perfectly innocent guy who got killed randomly. In a lot of ways his murder is one of the most disturbing ones in the story collection.
The professor may not really be a professor at all. He's a Russian man masquerading as a Englishman, so he could be pretending to be a real professor as well. At any rate, this guy is definitely one of the more unlikable characters in these stories. Coram betrayed his fellow revolutionaries in Russia and ran off to England, where he started a new life. He doesn't appear all that remorseful, or sorry, for his actions, even after his wife confronts him. And it's really his fault that his wife was there in the first place. Coram had documents that could free an innocent man from prison in Siberia. Coram does try to hide his wife from the police at least, but he doesn't really do much else that could be considered helpful or decent. He seems more caught up in his work and his own safety.
Anna is Professor Coram's wife and is the murderer of Willoughby Smith. Anna and her husband are both Russian and they were involved in a revolutionary group which she refers to as "Nihilists." Nihilism is a type of philosophy that celebrates "nothing": life has no intrinsic meaning, morals don't really exist, etc. In this case, Anna is probably using the term to refer to the Anarchists, a revolutionary political group that was active in late-nineteenth-century Russia. The Russian Anarchists wanted to overthrow the entire government and many engaged in terrorist acts and assassinations. Government crack-down on revolutionary groups was also quite violent in this period in Russia. To read more about the Russian Anarchists, check out this site, with lots of links and articles and this site, which, gives a rundown of the historical events leading up to the Russian 1905 Revolution.
Despite being a murderer, Anna is a highly sympathetic figure. She is basically blind without her glasses, and in the era before contacts and LASIK eye surgery, this was a real hardship. She also suffered time in prison in Russia and is married to a total jerk who abandoned her and their colleagues. Anna's quest is a noble one: she's trying to rescue a man named Alexis from jail. Anna ends up killing Willoughby in a panic, and after confessing everything to Holmes, she commits suicide by a poison capsule.
This is the man whom Anna was trying to rescue from jail. The fate of Alexis is why Anna broke into Coram's house, where she attempted to steal some documents that could free Alexis. Anna describes Alexis as a very close and loved friend, though there may have been more to that relationship. Anna never comes out and says directly. Fortunately, Holmes and Watson take the documents to the Russian Embassy as the story ends, and Alexis is presumably freed from jail in the future.
Wilson is the police officer who is at Professor Coram's house when Holmes arrives, and he answers Holmes's questions regarding the crime scene.
These people are all members of Professor Coram's household staff, and none of them really heard or saw anything that could help explain Willoughby's mysterious murder.