SOA , as an architectural style and a discipline of software engineering brings business and IT to the same table by providing a portfolio of business intended services/capabilities as the common terminology that he business requires and IT will design, implement and maintain. SOA is focused on establishing a center of operations for design around a portfolio of services, which equate to a set of business capabilities, that the business looks to deliver. Integration has become a strong focus for SOA as well. There are various types of services, integration and business services to name just two categories.
In the traditional Application Programming Interfaces or APIs were merely a library of interfaces that components can access and communicate with each other. In an evolved definition that is becoming more and more prevalent, the notion of a RESTful API that is exposed outside an organization, that may expose a set of services in the backend is being promoted more widely, especially with the proliferation of interest in HTTP-based simple interfaces that can mobile enable an enterprise's subset of back end functionality that it chooses to expose.
Cloud services are services that are exposed externally, and can be consumed by all, often implicitly assumed to be outside an organization's boundaries just as an API or more precisely, a RESTful API. Cloud services may span the gamut of Infrastructure as a service (IaaS) all the way to Software as a service (SaaS) which may expose very large grain applications or smaller levels of granularity akin to the RESTful APIs we discussed above.
More detail on this later.