What exactly is meant when asked to describe the author's characterization of characters in the novel?
What does it mean as in this question: What are some similarities in characterization in Gogol's Dead Souls and in Turgenenev's Fathers and Sons?
I do understand how to compare and contrast two characters
There is a difference between comparing the characters and the comparing the author's characterization. As you indicate knowing, comparing and contrasting characters covers such points as the personality; manner of behavior and speech; background including things like education, occupation, family etc.; goals, motives, objectives; physical appearance; any such things that might elaborate on the character(s) and show differences and similarities.
On the other hand, when discussing an author's characterization of characters, you'll be discussing such things as whether the characters are described through first person or third person point of view. You'll also discuss whether the author uses direct characterization (e.g., Alan Breck Stewart in Kidnapped) or indirect characterization (e.g., David Balfour in Kidnapped).
You'll also discuss more complex issues, if they apply, such as how the author creates the characters through literary structure, literary technique and genre-specific features. For instance, if it applies, you'll discuss whether the narrator gives a reliable or unreliable view of a character (e.g., Is the narrator biased or clear-sighted?); whether a character is actually reliable or unreliable in their own right regardless of other characters' opinions. If it applies, you'll discuss whether the characters display postmodernist fragmentation or metafictional awareness, and other features of these sorts.
In other words, you'll analyze and discuss the ways in which the author structures and creates the character, including whether symbolism or metaphor is attached to the character through their name (e.g., David Copperfield), hobby, occupation (e.g., Allan Quatermain), location, etc. So in Gogol's Dead Souls, you'll examine and analyze the text for indications of the structure, techniques and literary styles Gogol used:
To the door of an inn in the provincial town of N. there drew up a smart britchka—a light spring-carriage of the sort affected by bachelors, retired lieutenant-colonels, staff-captains, land-owners possessed of about a hundred souls, and, in short, all persons who rank as gentlemen of the intermediate category. In the britchka was seated such a gentleman—a man who, though not handsome, was not ill-favoured, not over-fat, and not over-thin.
For instance, after examining and analyzing the text, you might note that Gogol structures the story so as to have the third person narrator use a light and ironic tone in approaching the characterization of Chichikov. You might also note that Gogol devotes several paragraphs to describing Chicikov's actions, belongings, attitudes and appearance before ever revealing his name. Following the same kind of examination and analysis of Turgenev's text in Fathers and Sons, you might note that Bazarov is introduced amidst conversation and given a brief, to the point description:
'Daddy,' he said, 'let me introduce you to my great friend, Bazarov, about whom I have so often written to you. He has been so good as to promise to stay with us.'
Nikolai Petrovitch went back quickly, and going up to a tall man in a long, loose, rough coat with tassels, who had only just got out of the carriage, he warmly pressed the ungloved red hand, which the latter did not at once hold out to him.