Sometimes there's a problem with your iPhone, and a quick search shows that your best bet for fixing it is to restore your phone to factory settings and then restore from your backup. That used to be an Epic Quest, since even syncing to iTunes took forever and didn't always back up reliably. I recently had to swap out my usually maxed-out 32GB iPhone and I was dreading the restore process based on past experience. As it happens, I had nothing to worry about.
In iOS 7, restoring from backup can be paused, and then resumed later. So if it takes longer than you anticipated, you can always click the little X in the iTunes window and eject your iPhone. When you come back later, you can plug it back in, hit the sync button, and pick up where you left off. If you back up to iCloud, you have the same option, particularly handy if you inadvertently wander out of wifi range while you're in mid-restore.
First, make sure you're backing up regularly, whether to iTunes or iCloud. Within iTunes if you just want to back up and have no syncing to worry about, make sure your iPhone or iPad is plugged in, and select it in iTunes, and click the "Back Up Now" button on the summary screen.
Just above the 'Usage' bar in iTunes.
For iCloud manual backups, go to Settings > iCloud > Storage & Backup (at the bottom of the screen), then tap Back Up Now to kick it off.
Scroll down a skosh to see the backup options.
Luckily you have the option to cancel that iCloud backup too, and just like the restore, it will pick up where it left off when you go back and start it again.
Got a hot date? Cancel that backup until you have a free spot in your calendar.
I used to hate the backup and restore processes because they took forever, so I viewed them as a necessary evil. But both have been dramatically improved so now it's a lot less hassle to back up (which you should do regularly) and restore (which you shouldn't have to do very often). Knowing that it's not the Peter-Jackson-Film-level time suck it used to be makes it a lot easier for me to recommend when people ask for advice about how to fix This Weird Thing My iPhone Does Sometimes.