The Simple Object Access Protocol (SOAP) is a W3C (World Wide Web Consortium) recommendation that essentially allows for Remote Procedure Call (RPC) functionality over HTTP, via XML. (This is a simplification of the 120-page SOAP spec, but it suits our purposes). SOAP web services provides a systematic, defined way of communicating function requests and responses over a network transport.
SOAP interfaces are described by another W3C recommendation, WSDL documents - Web Services Definition Language. WSDL documents are the prototypes for SOAP functions. They define what parameters are expected to the functions, what formats are/aren't allowed, what will be returned, etc. Given the WSDL of a SOAP web service, programs can generate the client code that interacts with the services (as is demonstrated in the C# example project later).
Specifically for the Search Appliance, the SOAP interface provides, when using a language that has a SOAP API, a way to invoke a search and on the Search Appliance and insert data as if it were a local function call.