SOA web services means an approach to develop a web application that makes use of services. These services are often RESTful web APIs that return data back to the client. The service could also validate the client, or provide some type of analysis. SOA stands for service-oriented architecture.
A site that can book an airline ticket may also be able to book a hotel room. However, the credit card information is not being shared among different companies. Rather, the site is assembled with different meta-applications that provide various services.
These meta-applications act as services for the large application. The SOA web services approach is when you write an application that makes use of these services.
An API exposes functionality to the outside world through a set of industry accepted standards and protocols.