The different punctuation marks are used to distinguish quotes within quotes. Which ones go where really depends on where you are and who you're writing for.
The standard in America is to use double quotation marks except when placing a quote within a quote, as such:
"I was on my way home when I got a call from Peter," Maggie said. "I pulled over to answer the phone, and Peter said 'Maggie, I need to to pick a few things up at the store for me,' so I turned around and went back to the store."
In that text, everything except the words "Maggie said" was something Maggie was saying. But when Maggie quotes Peter, we indicate this with different punctuation. In different parts of the world, this rule is the same, but reversed. That is, in some countries, single quotes are standard except when writing quotes within quotes, in which case you use double quotes are used.