I like to think of Cloud Services as any service that moves the responsibility of data storage or computing to a set of servers operated by someone else. Cloud Services are often opaque, in that you as the user do not need to know much about how they are actually implemented, only that they provide a stable interface (API) that allow you to make use of these services.
Wikipedia defines “web service” as:
A web service is a method of communication between two electronic devices over the World Wide Web. A web service is a software function provided at a network address over the web or the cloud, it is a service that is “always on” as in the concept of utility computing.
Wikipedia defines “cloud computing” as:
Cloud computing is the use of computing resources (hardware and software) that are delivered as a service over a network (typically the Internet). The name comes from the use of a cloud-shaped symbol as an abstraction for the complex infrastructure it contains in system diagrams. Cloud computing entrusts remote services with a user’s data, software and computation.