Your iPhone stores old backed-up content in your iCloud. Unless you plan on rifling through months of app data and text messages, you don't need them. Clear old backups by following these steps: Settings > General > Storage & iCloud Usage > Manage Storage > Delete Backup. Now that you've removed past files, plug your phone into your computer and do a fresh backup via iTunes. Follow File > Devices > Back Up, and make sure you save photos and videos.
Your iPhone automatically backs up apps in your iCloud by default. If you don't want Snapchat or Words With Friends taking up your precious data, take them off the storage list. Go to Settings > General > Storage & iCloud Usage, tap Manage Storage under the iCloud section, select your device ("This iPhone"), and tap Show All Apps. You'll notice that Photo Library is taking up a lot of space. You've just backed up your photos to your computer, so turn that off for extra storage space.
There's a difference between iCloud and My Photo Stream. As the team at Thrillist put it, "The iCloud Photo Library is like paying to have all your photos stored in a safety deposit box, and My Photo Stream is like handing a secure copy of your photos to each of your dearest friends, and asking them to look after them for free." To set it up, go to Settings > iCloud > Photos, then turn off iCloud Photo Library and turn on My Photo Stream.
In three simple steps, you're done! You should now have enough free storage space to do a fresh backup (Settings > iCloud > Backup, and tap Back Up Now) and now all your precious information is safe. Easy.
For a detailed guide to managing your iPhone storage, visit Thrillist.
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