Application
Applying
noun, verbal noun
When you say humbly that you're requesting something, you're putting forward an application.
申請 refers to the process of officially applying for approval or permission, like ビザの申請 (application for a visa), ライセンスの申請 (application for a license), or 休暇の申請 (application for leave/vacation).
This is a jukugo word, which means on'yomi readings from the kanji. You already know this reading for 請, but not for 申, so here's a mnemonic to help you:
Today you finally submitted your application for a new shin (しん). Your old one had to be painstakingly removed after you banged it against a table. Anyway, congrats on the application for your new shin!
許可申請
application for permission
交付申請
application for issuance
登録申請
application registration
電子申請
electronic application
申請書
application form
申請者
applicant
申請手続き
application procedure
申請をする
to apply
申請を行う
to carry out an application
申請を受け付ける
to accept applications
申請が必要
an application is required
申請が通る
an application is approved
申請が受理される
an application is accepted
しまった！ビザの申請を忘れてた。
Oh no! I'd forgotten to apply for a visa.
申請手続きに時間がかかってるんです。
The application process is taking quite a while to complete.
ようやく許可申請が受理されました！
At last, the application for permission has been accepted!