To me...
An acquaintance is someone you see occasionally from the place you know them (work, school, etc), and your conversations are restricted to impersonal topics like the weather, current events, or work/school.
They become a regular friend once you start seeing them in your personal time and you like each other enough to do that. Or to a casual or workplace friend once you're comfortable talking more about your lives and being more honest with each other, even if the friendship won't last outside of work.
I have some trouble gauging when someone would like to move past chit-chat or considers me a friend as well.