'Leave' is time off from work, and is usually used when longer periods of time are taken off, such as when a baby is born or a family medical issue comes up. That time could be paid or unpaid. It is not, however, generally used when referring to vacation time. If you took a long weekend off, whether paid or unpaid, it would not be considered 'leave'.

'Leave' comes from the idea of permission. Any sort of leave means that you have applied for permission to not be at work. Some forms of leave are paid leave, some are unpaid leave. If you are away from work without leave, you can get fired. I disagree with what Carrie said about it being only for longer periods of time - but perhaps that is a local or a British/American difference. Typically we would not be given short periods of unpaid leave, as unpaid leave is complicated for your employer if you are on a salary, as they have to change your pay for that period. Normally you would be expected to use part of your allocation of paid leave. Note that it's vAcation, not vocation too.

Paternity, maternity and paid family leave are all typically paid leave. However, any time away from work is "leave". If you are not getting paid, it is often called "leave without pay" or "unpaid holidays".