Phone or iCloud storage is full
Your phone and your iCloud account are two separate stores. The fix depends on which one is full.
What's usually going on
"iPhone Storage Full" means the phone itself has run out of room. "iCloud Storage Full" means your online account has, which stops backups and photo syncing even if the phone has plenty of space. They are easy to confuse, and the warnings look alike.
Worth trying first
See which one it is.
For the phone, open Settings, General, iPhone Storage. For your account, open Settings, tap your name, then iCloud. Each shows what is taking the space.
On the phone, start with the biggest things.
The list is sorted by size. Old message attachments, downloaded movies, and unused apps are usually the easy wins.
Let the phone keep smaller photos.
In Settings, Photos, choose Optimize iPhone Storage. Full-size originals stay in iCloud, and the phone keeps lighter copies.
Remember that deleting a photo deletes it everywhere.
With iCloud Photos on, removing a photo from the phone removes it from iCloud and every other device too. Don't delete photos to make room unless you truly don't want them.
For iCloud, clear or upgrade.
Old device backups and large files are worth removing. If it's mostly photos, a larger iCloud plan usually costs less than the time spent pruning.
When to hand it over
If photos are spread across several phones, accounts, and old computers, or family members share one account. We'll get everything into one place, backed up, with room to spare.
Your own technologist, on call.
Tell us a little about your setup. We'll take it from there.