If you've got a smartphone or any kind of digital camera, you've probably got a ton of pictures built up someplace. And while organizing the mess would be an article unto itself, the most important thing is to make sure you're backing those pictures up. Merely moving them to an external drive or saving your SD cards isn't a great long term or disaster-proof solution. Services like BackBlaze or Carbonite will back up all of your data for a yearly fee (which is entirely worth it) but if your primary focus is on pictures, you can't have too many copies.
For a free photo backup that also gives you some amazing bells & whistles, Google Photo's is really hard to beat when it comes to features. All you need to have is a Google account (generally a Gmail address) and to install their free app. Once it's installed on your Mac, PC, iOS or Android device (or any combo), it will ask to search for, and back up, all the pictures it finds, automatically and for free.
One of the nice things that it does is combine all of your photos into a single, contiguous, library that you can then access on any device, anywhere, and easily share (or edit) those pictures to anyone you choose. By default all of the photos it uploads are only visible to your account but albums can be shared, and even collaborated on, via the apps and the web.
What's the catch?
Great question. The only catch (besides trusting Google) is that the free backup is restricted to 16 megapixel photo's. This should cover pretty much anything you would take with a typical smartphone or point-and-shoot camera. The more professional camera's, like DSLR's, create much higher resolution photos that would not back up at full size in this case. They do give you the option to back up at the highest quality but that might end up costing you a monthly fee if it fills up your account. Regardless of the limitations, it makes an excellent emergency copy if nothing else.
It's free so give it a try and see how you like it. Worst case scenario you can go into your Google dashboard and clear everything out!
Amazon Prime - If you're a subscriber to Amazon Prime ($99/year) you also get a side benefit of free photo backup, which is completely unlimited in size.
Flickr - Get 1 terabyte of free photo storage through this Yahoo-owned service.
Apple's iCloud - Definitely not free after a few gigabytes, but also an option for Apple users.