The "most important" kanji
There are tens of thousands of kanji in existence, but you don't need to know all of them in order to be considered fully literate. Below are the 3530 most important characters according to a few lists compiled by the Japanese government. The 281 atomic components were extracted from these characters.
Do I need to learn all of these?
The exact number of kanji you need to know really depends on how much you tend to read. If you're the kind of person who likes to read novels, you will apparently get by with about 3,500 characters; otherwise, you will probably get by with closer to 3,000 or fewer.
The kanji below happen to match 3,500 almost exactly. Combined, these four kanji lists offer the closest thing we have to an official collection of all the kanji you are likely to encounter in modern Japanese media aimed at the general public.
However...
A disclaimer
Lists like these should not be taken too seriously. I call them "the most important kanji" throughout Kanjijump only because it's too cumbersome to always type out "the closest approximation we learners have to an official collection of all the kanji you are likely to encounter in modern Japanese media aimed at the general public." In the end, "important" is a subjective term.
These lists were compiled at various times for various reasons, and none of those reasons was "to help the second-language learner of Japanese". That is to say, these lists may not be the best way to figure out which characters you personally should focus on learning next. That said, I reckon these lists serve as a good benchmark for your long-term goals as a student of Japanese.
The Education Kanji
教育漢字
The kyoiku kanji are those characters officially approved for use in the Japanese primary school curriculum.
This list was taken from the Japanese Ministry of Education
1006 characters
Grade 1
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寸干己至私穴我亡巻骨尺処孝論若片敬系貴供呼姿座泉並存異蔵郷従難段胸疑諸将背忘革降善尊晩城腹宗窓暮困派劇裏訳刻痛認衆探源就優視済射垂密権乱警専著盛危訪映宅枚樹純皇紅欲秘捨割厳値宣傷筋激頂否灰装層批吸幕忠延遺洗翌納展棒閉染砂聖勤推誌障党誤創模幼除看机俳宝拝簡律宇覧詞収乳策潮操針誠裁届討閣奮冊縦縮署熟暖仁奏脳宙拡絹臨補揮担臓后沿域賃蒸盟卵憲朗株郵穀庁糖肺誕班磁陛蚕鋼The Regular Use Kanji
常用漢字
The so-called jōyō kanji comprise those approved by the Japanese government for use in government documents and secondary school course materials. These are the only kanji you are likely to encounter in a newspaper without furigana phonetic marks. The jōyō kanji include all the kanji from the Education Kanji lists above, plus the additional 1,130 below, to form a total of 2,137 characters.
The list has been modified over the years, with some additions as well as removals. This list reflects the latest revision, from 2010.
1130 characters
The Kanji from the Extra-Regular-Use Kanji List
表外漢字字体表の漢字
In 2000, the Japanese government published a list of just over a thousand kanji falling outside the Regular Use Kanji list which they deemed to be common enough to warrant a standard form in print.
The list of 890 kanji given here includes all of those kanji, except those which have since been moved to the Regular Use kanji list (and are thus already listed above).
890 characters
The Personal-Name Use Kanji
人名用漢字
The jinmeiyō kanji are those approved for use in personal names despite falling outside the list of Regular Use Kanji. There are 863 in total, as of 2017; the list below includes just those 504 which are not also included in the Extra-Regular-Use Kanji List detailed above.
Many of the characters below are traditional forms (旧字体 kyūjitai) of characters in the Regular-Use Kanji list. That is, they are the original forms that were used before the characters were simplified to their present-day 新字体 shinjitai forms. Traditional variants of jōyō kanji are distinguished below via a double border.
504 characters