Your contract will give you the details of your paid leave, sick leave and time off in lieu. But, here is a quick guide to help you decipher that contract of yours.

National Holidays
JETs typically should not work on national holidays.
There are many national holidays during the year, and they should be listed within your contract and in your JET diary/calendar.
Note: days without official titles (like most national holidays) are still counted as 'national holidays' around the new year.
Silver week - two holidays which make long weekends in September
Golden week - group of holidays in early May which make long weekends. Note though, that to make your holidays last an actual week, you will usually have to take paid leave. Also note that at some schools, classes are still held during the two days in the middle of Golden week.
New Years holidays - For the new year in Japan, pretty much everyone gets about 3 days off, requiring no nenkyuu or other type of paid leave. During the NY holidays a lot of services (banks and gas/petrol stations) and attractions are closed for up to 5 days (usually until Jan 3). This can leave you without money, things to do, and without fuel for cars and heaters, so be wary.
This website has a guide to national holidays in the upcoming years: National holidays

JETs will sometimes be 'asked' or told to work on the weekend, for sports days, speech contests or other school events.
They may not be able to formally 'tell' you to work on these days, or Japanese beaurocracy will not have them outright demand you to work. But, if they ask you "Is it okay for you to work this Saturday?" that normally means you should be working this Saturday. This leads to time off in lieu.

Time off in lieu
Called 'furikyuu' or 'daikyuu', it is time off you get for working on a day that should otherwise have been a holiday.
Sometimes, this day is chosen and all the staff and students will not come to school that day. Other times, you will be able to choose a day to take off. Often it is within a certain amount of weeks of the 'extra work day'.

No classes
For example, summer break/spring break/winter break.
When you see these 'no classes' days written on the yearly schedule, you would normally think 'no school!'.
Generally in Japan, except national holidays, everyday is a 'school' day. So, you will (probably) be expected to be at school or your BOE during summer/winter/spring breaks. Teachers and students often practice club activities during school breaks.
Some JETs are not expected to work on these days, so you will know for sure by asking your workplace/supervisors.
If your weekly schedule has no classes planned, you should most probably still be at school working for your contracted hours. Ask beforehand if you are unsure.

If you are late, they may take some of your annual paid leave in hourly increments.

Annual paid leave - days off
Often called 'nenkyuu'.
The number of paid leave days varies for each JET.
Also, the way you use the paid leave and when you can use it can vary.
Strict supervisors/BOEs may only let you use your paid leave during the 'no school' breaks.
Other schools may rearrange the work schedule for you to take leave during normal working days.
Often, you should ask your supervisor directly when you want to use your paid leave days.
It is recommended to use your paid leave on days when you have no classes, such as during winter/spring/summer break, or during exam periods. However, this is not an obligation, and it is up to your supervisors discretion when you can and cannot take leave.
It is usually not acceptable to try to take paid leave when you have a visit to a school that you go to infrequently, like only once or twice per term.

Annual paid leave - hourly
Your contract will probably stipulate that you can take paid leave in hourly increments. So, you can take paid leave for part of a day, and the hours will be deducted from your total paid leave days.
For example, if you work 7 hours per day, and take an afternoon off (3 hours), you will have 4 hours of that day of paid leave left.
This is often utilized by JETs who have to visit a doctor or run errands during work hours.
However, some schools may not let you leave at all during work hours, and others may not ask you to take paid leave for visiting a doctor, bank etc. during work hours.

Extra leave after recontracting for another year
Some boards of education will give extra days of annual paid leave for the next contract year.
Also, sometimes during the summer break you will be given an extra number of 'free' days that are not annual leave days. These are often only allowed to be used within a certain time frame, or the summer break itself, otherwise they expire.

Sick leave
Sick leave is called 'byoukyuu'.
Taking sick leave can be tricky in some situations. Sometimes, your supervisors/BOE may try to deduct annual paid leave days, instead of sick days.
Often, it is necessary to visit a doctor and obtain a doctor's certificate in order to take sick leave.
Your contract will stipulate how and went to take sick leave, and also how many days you can take sick leave consecutively.

Special leave - emergencies/problems
If you have special circumstances, like a death in your family, problems back home and so on, you need to talk with your supervisors about your options.
Your contract should list types of emergencies, and how many days leave you get.
Ultimately, it is up to your supervisors/BOE to decide if you should be given special leave, and you should talk to them directly if you ever have a serious problem.