1To be or get involved in something — to take part in an activity or matter, or to interfere in it.
彼は地域の防災活動に関わっている。
He is involved in local disaster-prevention activities.
他人のけんかには関わらないほうがいい。
It's better not to get involved in other people's fights.
事件に関わったとして数人が事情を聞かれた。
Several people were questioned for being involved in the incident.
2To relate to or have a connection with a matter — often used adjectivally as "a matter relating to ~."
この報告書は税制改革に関わる内容を含んでいる。
This report contains content related to tax reform.
個人情報に関わる規則は年々厳しくなっている。
Regulations relating to personal information are getting stricter every year.
3To have a serious bearing on something — used especially in phrases like "life-threatening" (命に関わる) or "affecting one's future" (将来に関わる), where the stakes are major.
その病気は命に関わることもある。
That illness can be life-threatening.
この決定は会社の将来に関わる重要な問題だ。
This decision is an important matter that affects the company's future.
4Written with the kanji 拘る: to stick to or be particular about something, paying close attention to detail — close in meaning to こだわる.
彼は素材の質に拘って商品を作っている。
He makes his products with strong attention to the quality of materials.
細かい部分に拘りすぎて作業が進まない。
He's so particular about the small details that the work isn't progressing.