If you find the characters in a novel sympathetic, it means you identify with them, you understand why they act as they do, you are interested and concerned about what happens to them.
In the case of Pride and Prejudice, how do you feel toward Elizabeth Bennet? Do you admire her ability to ability to analyze the good and bad points about the people around her, or do you think she is opinionated and unwilling to learn from others? When she says to Jane,
Affectation of candour is common enough — one meets with it everywhere. But to be candid without ostentation or design — to take the good of everybody's character and make it still better, and say nothing of the bad — belongs to you alone.
is she complimenting Jane for her ability to accept others as they are, or is she making fun of Jane's gullibility? If you feel she is being kind in her comment, you are probably sympathetic toward Elizabeth.
To find a character in literature "sympathetic" means that the reader can identify with and/or understand the motivation and conflicts of that character. The reader sympathizes with characters when we recognize their struggles as real, their strengths and weaknesses as those encountered by humans in general.
For example, in The Great Gatsby, Jay Gatsby's weaknesses such as his naive idealism and attachment to a faded dream, as well as his strenths, such as ambition and devotion, evoke sympathy from the reader who can relate to those motives. Without the narrator Nick's perspective in revealing Gatsby's traits to the reader, though, Jay Gatsby may not be so sympathetic a character.
To understand whether a character is sympathetic, the reader should be able to identify which character traits evoke sympathy from the reader, and be able to cite specific passages that especially reveal the sympathetic qualities of the character.