If Prince Charles dies before Queen Elizabeth who will succeed to the throne? Prince Andrew? If so, after him, would his children be next in line for the throne?
Prince Andrew, younger brother of Prince Charles, is currently fourth in the line of succession; the primary line goes through Prince Charles and his descendants, making Charles's oldest son, Prince William, the next in line, should Charles die before Queen Elizabeth. William is followed by his younger brother Prince Henry, at least as long as William remains unmarried and childless. Then comes Prince Andrew, followed by Prince Andrew's children (5th and 6th place), followed by Andrew's younger brother Prince Edward, followed by Edward's daughter, followed by Charles's sister, Princess Anne, followed by her children.
If William has children within the bounds of wedlock, those children would follow him in the order of succession; they would then be followed by Henry, Henry's children (if any), and only then Prince Andrew.
In short, the line of succession goes to the first person in line, then that person's descendants; only after exhausting one branch does one go back up the family tree and down the next path. Within any given set of siblings, sons have precedence over daughters, and the elder has precedence over the younger.
You'll probably want to check out the line of succession on the official website of the Royal Family; you might cross-reference that with our royal family tree. We also have a complete list of the rulers of England and Great Britain.