What's the difference between a "sick day", a "personal day" and "vacation time?

In my old job, part-time workers (like me) didn't get any benefits, and if you wanted a day off you had to warn 16 days in advance and you didn't get paid for that day. If you have an unexpected trouble you can call out but you have to have a reason. At my new job, part-time employees have stuff like sick days and personal days and vacation time but it's all really confusing. We have to alert the managers a month in advance, but does that mean you have to notify them a month in advance if you want to use a sick or personal day? And what's the difference between vacation time, personal and sick? Please help me because I'm so confused.