Here are some cloud databases with REST API services that make building a web / mobile app with a backend database super easy.
Apigee Usergrid
1,000,000 requests per hour for FREE
Kinvey
1,000,000 requests per month for FREE
MongoLab
https://mongolab.com
240MB FREE
StackMob
60,000 requests per month for FREE
Parse
https://parse.com
1,000,000 requests per month for FREE
Google Cloud Storage
Not free
I went with Parse because it was the easiest to use.