Learn Hardcore Korean: jungyohan sajineul hyudaepone jeojanghae duseyo.

In Korean, -을/를 marks the direct object of a verb. Here, 사진을 shows that the photo is what you’re saving.
-에 indicates a location or destination. So 휴대폰에 means “in/on your phone” – that’s where you’re saving the photo.

What does 저장해 두세요 mean? Why not just 저장하세요?

저장하세요 simply means “please save.” Adding -어 두다 (which becomes 저장해 두다) adds the nuance of doing something in advance and keeping it that way.
So 저장해 두세요 is like saying “please save it now and have it ready (for later).”

How does 중요한 work? Why not 중요하다 사진?

중요한 is the adjectival form of the verb/adjective 중요하다 (“to be important”).
In Korean, you turn -하다 adjectives into modifiers by changing -하다 to -한 (for “adjectival” usage).
So 중요한 사진 means “important photo.” You can’t place 중요하다 directly before a noun without this change.

Why is there no subject in the sentence?

Korean often omits the subject when it’s clear from context. Here, it’s understood you (당신 or 너) are the one being asked to save the photo. Imperatives in Korean typically drop the subject.

What level of politeness is -세요?

-세요 is the polite, informal/formal polite ending. It’s suitable for speaking to people you’re not super close with (e.g., co-workers, acquaintances, customers). It’s more polite than plain -해, but less formal than -하십시오.

Could I use other verbs for “save,” like 보관하다?

Yes, 보관하다 means “to store” or “to keep,” but it often implies longer-term or physical storage (e.g., storing files in an archive). 저장하다 is specifically used for saving digital data. For photos on your phone, 저장하다 is the most natural choice.

Can I shorten 저장해 두세요 in casual speech?

Yes. In very casual speech among friends, you might hear 저장해 둬 or even 저장해 놔.
But be careful: casual forms drop politeness and are only appropriate with close friends or younger people.