について details

Examples

about N; regarding N — the topic of speech, thought, or writing

週末しゅうまつの予定よていについて、もう聞きいた？

shuumatsu no yotei ni tsuite, mou kiita?

Have you already asked about the weekend plans? casual

このレポートは地球ちきゅう温暖化おんだんかについて書かかれている。

kono repooto wa chikyuu ondanka ni tsuite kakarete iru.

This report is about global warming. neutral

本日ほんじつは新しんサービスについてご説明せつめいいたします。

honjitsu wa shinsaabisu ni tsuite go setsumei itashimasu.

Today I will explain our new service. formal

Modifying a following noun needs の: 経済けいざいについての本 "a book about economics" — bare について本 is not grammatical.

に関して
Both mark the topic under discussion, but について is the everyday default in both speech and writing, while に関して skews formal/written even once the surrounding sentence turns casual — 週末について聞いてもいい？ reads unmarked, its に関して twin still reads a shade stiffer even in casual conversation.

に対して
Both take the family's に＋[verb]て shape and a dictionary flattens both to "about/regarding," but について names a neutral topic of speech or thought while に対して marks the target of an action or attitude directed AT the noun — 日本について話す (talk about Japan) has no に対して equivalent, and 彼に対して怒る cannot become ×彼について怒る.
において
に「おいて」and に「ついて」differ by one mora and both gloss loosely as "in/about" in casual translation, but において marks a formal domain, venue, or point in time an event occurs within, while について marks the topic of speech or thought — 現代社会において起きている (is happening in modern society) is not interchangeable with 現代社会について起きている, which is not idiomatic Japanese.
につれて
One kana apart (れ vs い) and easy to misscan at reading speed, but につれて demands a gradual proportional change between two nouns or clauses, while について just names a topic — 発展につれて (as it develops) is not the same word as 発展について (about the development).