Cloud storage is great for documents where you need to be able to access the same file across different devices. This also comes in handy for photos, since cloud storage enables you to have whatever photos you like on whatever screen you like. This can mean your computer, your iPad, or even as the photo screensaver on your Apple TV. You don't have to create that special folder anymore to sync photos back to your devices and hope you got them copied everyplace, as long as you have an internet connection they're available.
Bear in mind that cloud storage might not even be for you. Bryan Chaffin and John Martellaro, aren't using any sort of cloud photo storage, while Jeff Gamet and I are all local storage but dabbling in Photo Stream (Jeff's a heavy user and I only use it when I want to share images with other people).
While the default choice (because it's what shows up automatically) would be iCloud Photo Library, there are a variety of options available for cloud storage (and sharing) of photos, all at varying price points and with slightly different features.
Here at TMO, we have two people using iCloud Photo Library: Adam Christianson and Vern Seward. Both of them back up to other places, but are using iCloud Photo Library as their primary storage. Dave Hamilton sort of uses it, but mostly uses a combination of other services, like a Transporter and BitTorrent Sync to keep things available wherever he happens to be.
Here's iCloud Photo Library showing all the faces in your albums.
On Thursday at Google I/O, Google Photos was announced, with free unlimited storage. It's available right now for Android and iOS devices, and also via web page. So far there aren't a lot of details about the terms of the service aside from a 16 megapixel limit on images, but since Google is known for their unlimited storage space it might be worth a closer look.
Aside from Google, there are other "cloud storage that also has photos" providers. Amazon, Microsoft, and Dropbox all have photo management built into cloud storage solutions that might be useful. All of them offer a certain amount of free storage or a trial that varies depending on how you got signed up. Beyond the initial free space you have an option to purchase additional storage, so on the upside these are services that can scale as needed. Also because these aren't specific to photos, once you're signed up and set up, you can sync whatever you want and have documents or other images or audio or video files available no matter where you are or what device you're using.
Cloud storage is also available from SmugMug and Flickr, which are specific to photo storage. Flickr offers a terabyte of storage for free, while SmugMug charges US $60/yr for it's lowest priced plan. One of the advantages to charging for everything is that SmugMug has been around for 13 years and shows no sign of going anyplace. Flickr still shows promise, but it gets harder to hold out hope as the current offering doesn't seem to evolve at all and it's confusing what you can do in the app vs on a mobile web page vs on a computer web page.
You can tell how much I use my Flickr account...
All of these services offer auto-upload on iOS, so getting your pictures there is easy. Aside from Amazon, all of these services offer a Web Gallery feature, so you can share with others. Aside from Google, all the others offer sharing to external social networks, so you can still share to Instagram or Facebook if you want to.
If you are interested in using cloud storage for your photos, keep in mind you'll still need to back them up someplace else, but you can use that cloud storage to make them more readily available on all your devices. Aside from that, cloud storage is becoming a more attractive option all the time, both in price and convenience. A bit of research on the one that sounds the most appealing will help make it clear which one is the best fit for your particular setup.