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 278 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.
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
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
The Regular Use Kanji
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 2,136 characters.
The list has been modified over the years, with some additions as well as removals. This list reflects the latest revision, from 2010.
The Kanji from the Extra-Regular-Use Kanji List
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).