Welcome to the Grammar Minute, where we’re saving the English language sixty seconds at a time! I’m Lauren Smyth, and I’m here to talk about quotation marks in nested quotes. By this I mean quotes inside quotes, something like: He said, “I asked her if she was going home, and she told me, ‘I’ll head home after lunch.’”
The first thing to know about nested quotes is that they should be avoided. They’re clumsy and hard to follow. But if you must have one, the rule for quotation marks is that the first, outer marks should be double quotes. The next, inner quotation marks should be single quotation marks. If, heaven forbid, you have a third nested quote, switch back to double. 
I’ve never seen a triple nested quote, however, and I hope I don’t, as it’s extremely confusing for the reader.
The bottom line is: Alternate quote styles, starting with double quotes and switching to single quotes for the nested quote. If both quotes end at the same point, be sure to type out and close both sets of quotation marks.
That’s your Grammar Minute! Visit thegrammarminute.com for more tips and tricks.