Great overview of some current cloud services with APIs for mobile and web apps:
DevOps to NoOps: 10 Cloud Services You Should be Using (Russ Mason)
Some cloud services mentioned in the presentation to check out:
SendGrid (sending email)
PubNub (fast messaging)
Twillio (telphony services)
Russ Mason is founder of MuleSoft. They provide a layer between a bunch of the APIs and Apps.