The locations of most interest are the 3 courts in Shiba Koen (Park), the 4 in Hibiya Koen, and the 48 (!) in Ariake. There are also 14 courts in Oi, a bit of a ways south past Shinagawa; and 6 in Kiba which is east of the Ginza area a ways. See the pages for each of these locations. Tokyo Metro also has courts in the western, eastern, and northern parts of Tokyo - a total of 206 courts in all.
At 1800 yen per hour on week-ends and holidays, the money cost of these courts is quite low - offset by your time and effort to go through the lottery. You'll find it useful to band together with a group of like-minded friends who also enter the lottery and pool the results (see the page on "cooperatives").
To join the monthly lottery, you first have to register. You can do this at the admin offices at each location. For Shiba, the admin office is on the other side of the baseball field from the tennis courts, at the corner of the large intersection. For Hibiya, the admin office is right next to the tennis courts. You must register in person. You'll need either your alien registration or your Japanese drivers license. The staff will help you fill out the form. You will be given a registration card that has your ID number. You will use this number and your birthdate in the lottery. The card is valid for 3 years. After registration, you'll have to wait about a week before you can use it, due to the time needed to get your registration info into the system.
The lottery is done in a monthly cycle. Each registered person can apply for two specific 2-hour time slots per month. An independent "drawing" for each time slot is conducted among those persons who applied for that specific time slot. For example, let's say you and 20 others applied for 9 to 11 AM on October 9, a Saturday. The drawing for that slot is done among just you and those 20 others. If for another slot, say 7 to 9 PM on October 10, a Monday, you and only 5 others applied, then the drawing for that slot is done among just you and those 5 others. Obviously the more popular times will have more applicants and your chances of hitting will be lower, so banding together with your friends is very very helpful in improving your odds.
Let's go through a monthly cycle. You turn in your lottery request (see below for methods) between the 1st and the 10th of the month. For October this will be between Sept 1 and 10. The lottery is conducted on the 13th. If you use the postcard method of entering the lottery, you will be notified by postcard of the results. If you use the telephone to enter the lottery, you must check results by phone between the 15th and 20th of the month, i.e between Sept 15 and 20 for the October cycle. If you don't check within that period your entry will be nullified even if you won.
When you "hit" in the lottery, you must go to the admin office to pay for your court time just before you play. Bring your registration card with the ID number - you may be asked to show it.
There are 3 ways to enter the lottery: by computer over the internet (recommended), by phone, and by a computer terminal at the admin office. All of these are in Japanese.
The computer/internet is the most straightforward, and you can do it anywhere that you have an internet connection. The internet location is www.sports.metro.tokyo.jp. The screens are all in Japanese. You can also check on still-untaken courts at this site.
You can also use a terminal that is at each admin office. These get crowded at peak periods.
In the telephone method, you make responses using your phone's keypad. This is rather tedious but if you don't have an internet connection it is doable. Again, all in Japanese. You'll need a chart that shows the various codes for the locations, times, etc etc; get this chart at an admin office when you register for the ID card. Call 3371 5489 to use this method.
If you have to cancel your use of an obtained time slot, be sure to do this at least 2 days before the day of use, using oneof the methods above. If you simply don't show up, or cancel too late, the system will keep track of how many times you've done that and at a total of 3 occurrences you will be barred from entering the lottery for 3 months. After that, if again you just don't show up or cancel too late, your registration card will be cancelled and you cannot re-register for 1 year.