To take on, undertake, or accept responsibility for a task; to take charge of something.
彼は新しいプロジェクトを引き受けた。
He took on the new project.
忙しかったが、手伝いを引き受けることにした。
Although I was busy, I decided to agree to help.
部長が重要な仕事を引き受けてくれたので安心した。
I felt relieved because the manager took on the important work.
To take over or inherit duties/roles (to assume someone else's responsibilities).
父の店を引き受けることになった。
I ended up taking over my father's shop.
来月から彼が私の業務を引き受ける。
From next month he will take over my duties.
To guarantee or vouch for something; to assume responsibility such as acting as guarantor.
彼は保証人として借金を引き受けた。
He agreed to take on the debt as guarantor.
会社が製品の品質を引き受けると約束した。
The company promised to guarantee the product's quality.
To contract (an illness) — a rarer usage meaning to take on or catch a disease.
出張中に風邪を引き受けてしまったようだ。
It seems I contracted a cold during my business trip.