What is meant by the expressions "rules of inheritance" and "gender bias"? How do they apply to The Lion, the Witch and the Wardrobe?

"Gender bias" refers to treating people unfairly because they belong to a certain gender; traditionally, women and girls are given less credibility than boys and men (though this type of bias is starting to change).
In The Lion, the Witch, and the Wardrobe, Lucy, the youngest child and a girl, finds Narnia first. After she returns from the magical world that she finds by passing through the wardrobe, the others hardly notice that she was missing. She tells her older siblings, a sister and two brothers, "But I've been away for hours and hours." Her older brother, Edmund, says, "Batty!" He then taps his head and says, "Quite batty." Her sister, Susan, says, "Don't be silly, Lucy.... We've only just come out of that room a moment ago, and you were there then." And Lucy's older brother, Peter, says, "She's not being silly at all...she's just making up a story for fun, aren't you, Lu? And why shouldn't she?" No one believes Lucy, perhaps in part because she is the youngest and a girl, and this situation makes Lucy quite miserable: 

"But Lucy was a very truthful girl and she knew that she was really in the right; and she could not bring herself to say this. The others who thought she was telling a lie, and a silly lie too, made her very unhappy."

Lucy lives in a state of misery because no one believes her, in part because of her gender and in part because of her age. Her siblings consider little girls prone to making up stories. Edmund then enters the wardrobe and sees that Narnia is a reality; only then does he believe Lucy. However, when he leaves Narnia, he lies and tells Susan and Peter that Lucy imagined Narnia to play a cruel trick on Lucy.
The "rules of inheritance" define who can inherit titles and property when someone dies. In The Lion, the Witch and the Wardrobe, the White Queen tells Edmund that she will bring him up to be a prince. She tells him:

"You are to be the Prince and--later on--the King; that is understood. But you must have courtiers and nobles. I will make your brother a Duke and your sisters Duchesses."

In reality, the Queen's election of Edmund to be king goes against traditional laws of inheritance, as the oldest boy, Peter, would be king before Edmund. According to traditional rules of inheritance, boys inherited the throne before girls did, which also shows gender bias.