The KANKEN is a Kanji Test administered by the Nippon Kanji Kyoiku Shinkokai.
It is not just one test, but a dozen tests divided into levels. You can take any level you want.
Level 10 tests the 80 Kanji taught in the first grade of elementary school as well as kana.
Level 9 tests 160 Kanji from the second grade of school. Of course, each level may include Kanji from the previous levels, so you need to know 240 Kanji for this level. Levels 10 and 9 are not available during the open public testing days. Private groups can offer these levels, but for most of us on this forum, we had better start with level 8.
Level 8 covers a total of 440 Kanji. You need to know how to read and write the characters. You also need to know the correct stroke order and number of strokes. Reading is tested by sentences which have the Kanji in them and you have to write the reading of select words in hiragana. Writing the characters is also required. Another section will have sentences and instead of Kanji, the tested words will be in hiragana. You have to write the Kanji. Stroke order is tested by showing a character with one of the strokes in bold. You have to write the order number of that stroke. You also need to know antonyms. One word is given to you and you have to write the missing Kanji of the other word which is the opposite meaning of the first word. Then there is a section where two words have a Kanji which has a shared component. You have to write both characters. Another section tests you on writing Kanji which have the same reading. Each Kanji is in a sentence and the reading is given to you. You write the Kanji. You should also know the correct okurigana. That is the hiragana used after a Kanji as part of the same word. Another thing you should be aware of is the character radical.
Level 7 covers a total of 640 Kanji. For this level, besides reading, writing, stroke order, radicals, okurigana, and antonyms, it includes 3-character Kanji compounds. You must also be familiar with Kanji that have the same reading. On-yomi and Kun-yomi are tested. There's also a note about being careful with your strokes.
Level 6 covers a total of 825 Kanji. Reading, writing, antonyms, synonyms, homonyms, 3-character compounds, okurigana, stroke order, stroke count, radicals and the name of the radical, relationship of characters in a compound word.
Perhaps by now you can understand how the test gets more difficult as you progress higher so I think I'll just list the total number of characters for the rest of the levels.
Level 5 - 1006 Kanji. These are all the characters learned in elementary school (grades 1 through 6). Includes 4-character Kanji compounds.
Level 4 - 1322 Kanji. Includes ateji which are words that have a reading not based on the readings of the Kanji.
Level 3 - 1608 Kanji.
Level Pre-2 - 1945 Kanji. This amount covers all the government-approved Kanji for daily use.
Level 2 adds 285 Kanji approved for use in personal names.
Level Pre-1 - 3000 Kanji.
Level 1 - 6000 Kanji. That is six-thousand Kanji! And you thought the jump from the JLPT level 2 to 1 was steep.
The thing that makes this test difficult for foreigners is actually the vocabulary. You have to know more than just how to write and read Kanji. You have to be able to come up with the characters for words that may not be so common or useful to you. What do you think of when you read とうだい? If you thought of a famous university, you would be wrong. You see, you've got to know these words even though you'll never use them. It can be easy to come up with nothing. Even though you can write every single Kanji required and you know the meanings of those Kanji, it simply won't be enough. You need vocabulary. If you study for this test, you will need to learn a lot of words. How you are going to remember them all, I have no idea. Maybe you should make up a story with your new words. I don't know of an official word list. It probably doesn't exist.
There are at least a few publishers who create study books for each level of the test. Perhaps the most famous series is the one published by the same group who runs the test. These are called Kanji Gakushu Step and go from level 10 to level 2.
The last 2 levels are in their own series called Kanzen Seihuku (完全征服)
Level Pre-1 is this book:
Level 1 is this book: