To apply for, request, or formally sign up for something; to make an application, reservation, proposal, or offer (including marriage proposals).
大学に奨学金を申し込んだ。
I applied for a scholarship at the university.
旅行会社にツアーを申し込みたいのですが、空きはありますか？
I'd like to book a tour with your agency; is there availability?
彼は彼女に結婚を申し込んだ。
He proposed marriage to her.
私は会社の研修会に参加を申し込みました。
I signed up to attend the company's training session.