申请(shēn qǐng) - to apply for, to request formally; an application in Mandarin

申请 (shēn qǐng) is the formal word for applying: for a job, a school place, a visa, a licence, a grant. It implies paperwork and an institution on the other end, so it is not the word for asking a friend for a favour, which is 请 or 求.

It works as both verb and noun with no change of form. 我申请了 is I applied; 我的申请 is my application. The paperwork itself is 申请表 (application form) and the applicant is 申请人.

The English preposition disappears. You apply for a job, but in Mandarin 申请 takes the object straight: 申请工作, 申请大学, 申请签证. Do not reach for 为 or 给 to translate "for", and note that the tone pair shēn qǐng is first then third, so the second syllable dips low.