If someone is self-centered, it means they are generally quite selfish and only do things that benefits them. To also explain this we say someone only cares about themselves. It is a negative phrase.
"self-centered" は、身勝手（selfish）で自分の[ためになる](https://eikaiwa.dmm.com/uknow/questions/65390/)ことしかしない人を表します。
"only care about oneself" も同じ意味です。ネガティブなフレーズです。
When someone is selfish, it means that they only care about themselves and not others. We have lots of ways to say the word selfish.
egocentric
egotistical
self absorbed
self centered
"He has thoughts for no one but himself."
All of his thoughts are focused on himself. He does not think about anyone else.
Other examples:
"He thinks of no one but himself."
"He is only concerned with himself." (Concerned - to think about his needs, worries, etc.)
"He has thoughts for no one but himself."
（彼は他の人はどうでもいいけど、自分の事ばかり考えています）
彼は自分の事ばかり考えているという事になります。
他の人の事は全く考えない、と言う事になります。
【その他の表現】
"He thinks of no one but himself."
（彼は他はどうでもいいけど、自分の事ばかり考えます）
"He is only concerned with himself."
（彼は自分の事ばかり気にしています）
(Concerned - 自分に必要な事、心配事など.)
Only caring about one's well being means that he does not care or have compassion for others or their issues.
Being absorbed in one's own affairs, implies that they are too busy thinking about their own life to care about anyone else.
「only caring about one's well being」は、他の人のことを気にかけない、思いやりがないことを表します。
「being absorbed in one's own affairs」は、自分のことで頭がいっぱいで、他の人のことを考えられないといった意味合いです。
Self centred ' describes someone who believes the world revolves around them. Other people are either unimportant or, at best, secondary. An alternative may be 'self obsessed'.
A 'conceited' person is one who believes that no one else matters but himself. This is a person who is extremely proud of himself/herself that he believes everyone else does not matter. This type of person is self-centered. You may also call him/her a selfish person because he/she has no consideration for other people but is concerned only with his/her own personal profit or pleasure.
So, you may say:
The man is so conceited that he could not let anyone else speak in the meeting.
or
Have you noticed just how selfish she is? She always wants everything for herself.
「conceited（うぬぼれの強い）」は、自分以外の人はどうでもいいと考える人を表します。ものすごく自尊心が強い、自分が一番偉いと思っている人です。このタイプの人は、「self-centered（自己中心的）」です。「selfish（利己的）」とも言えます。他人のことを考えず、個人的な利益や楽しみにしか関心がないからです。
下記のように言えます。
The man is so conceited that he could not let anyone else speak in the meeting.
〔訳〕その男性はとてもうぬぼれ強いです。彼は会議中、他の人に話させようとしなかったです。
Have you noticed just how selfish she is? She always wants everything for herself.
〔訳〕彼女がどのぐらい利己的か気づきましたか？彼女はいつも自分の思い通りにしたがる。
1.
The hyphenated word "self-centred" expresses that the focus is always on that person. In other words, this person only cares and thinks of himself/herself
2.
Same as above.
3.
This expression is a metaphor to show that if the world revolves around something, it is the most important thing in the world.
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