Use double quotes for a quotation.
Joe said, "Will you marry me?"
Use single quotes for a nested quotation, when someone repeats what someone else said.
Joe smiled and said, "Jenny said 'yes' when I asked her to marry me."
If you need another layer of quotation, just keep alternating between single and double quotation marks.
"Joe was just here," said Susan. "He said, 'Jenny said "yes" when I asked her to marry me.'"