Full-time employment means that you are paid a monthly salary for a given amount of teaching hours per month. This salary should be paid despite public holidays, school activities, and other random class cancellations.
If you are teaching at a high school, full-time may also mean having to be at school for assembly, and having to join in extra-curricular activities. Full-time jobs may also bar you from doing outside work during school hours even if you don't have any classes. Despite the full-time label, some schools will not offer 12-month contracts, i.e. you will not get paid during the summer months. Universities generally allow full-time teachers to come and go as they please, and usually give 12-month contracts.
Part-time work could also be described as freelancing. Part-time work is paid per class taught at an hourly rate. The upside is that the teacher only has commitments to the classes he/she teaches and not to any other aspect of school life, with the teacher working as little or as much as he/she wants. The downside of course is that a part-time wage is not very stable considering the amount of public holidays, and class cancellations that are guaranteed to occur throughout the year. Overall, part-time work is good for newbies and retirees in need of something to occupy their time.