Soap, REST, xmlrpc. Facebook, twitter, [insert web 2.0 site]. What is the definitive web API and which would be the one that you would most likely replicate in your own code and for what reason? It seems that some web APIs invoke nausea in developers while other invoke pure worship. If you were told to develop an API for a website/product which would you emulate (even if it was overkill)?
Products from 37signals have great APIs, using RESTful XML-over-HTTP. The documentation is reasonably good, too.
The Google GDATA set of APIs - I use the YouTube one - extends Atom, which is a standard in its own right, so time spent developing with it is also time spent learning about this RSS-type protocol, which gives it an edge over more proprietary ones IMHO.
Flickr's API is pretty good: http://www.flickr.com/services/api/
Spent a few months working with it and found it pretty usable. The docs are great too, which many have made it seem that much easier.
REST, JSON/XML/PHP/SOAP protocols, every endpoint is namespaced, lots of options/parameters for every request. Error messages and codes are documented too.
Most of all it lets you access just about all information Flickr stores about its photos and users (with appropriate permissions, of course). Gotta love good APIs!