An API, or application programming interface, defines a set of classes, functions, and structures to be used by an application to make use of some library or subsystem. For example, both the windows multimedia subsystem, and windows sockets subsystem both have their own API. An API is not a concrete entity, you can't point at a file and say that the file itself is an API. An API is merely a specification for a communications protocol that a program needs to use to make use of a library or subsystem.
An SDK, or software development kit, contains tools, documentation, and needed files, to program against 1 or more APIs. Some SDKs, but by no means all, may contain sample code to demonstrate how an API can be used. 2 examples of an SDK are the Windows Platform SDK and the .NET Framework SDK.
The most likely reason the terms are used interchangeably is because sometimes an SDK only has the tools, documentation, and files for a single API, and both the API and SDK share the same name. An example of this would be the SDK for developing winamp plugins.