REST (Representational State Transfer) and HATEOAS (Hypermedia as the Engine of Application State). It seems like everyone claims to be RESTful, but mentioning HATEOAS is frequently met with silence. Of course, these terms come from Roy Fielding's PhD dissertation, but I won't claim that it is very readable (it is not the nature of dissertations to be readable...). Fortunately he's provided more readable blog posts about REST and HATEOAS. At the risk of aggressively over-simplifying things, REST = "URIs are nouns, not verbs" and HATEOAS = "follow your nose".
"Follow your nose" simply means that when a client dereferences a URI, the entity that is returned is responsible for providing a set of links that allows the user agent to transition to the next state. This standard procedure in HTML: you follow links to guide you through an online transaction (e.g., ordering a book from Amazon) all the time -- it is so obvious you don't even think about it. Unfortunately, we don't hold our APIs to the same level; non-human user agents are expected to make state transitions based on all kinds of out-of-band information encoded in the applications. When there is a change in the states and the transitions between them, there is no direct communication with the application and it simply breaks.
I won't dwell on REST because most APIs get the noun/verb thing right, but we seem to be losing ground on HATEOAS (and technically, HATEOAS is a constraint on REST and not something "in addition to REST", but I'm not here to be the REST purity police).
There are probably many good descriptions of HATEOAS and I apologize if I've left your favorite out, but these are the two that I use in my Web Server Design course (RESTful design isn't goal of the course, but more like a side benefit). Yes, you could read about a book about REST, but these two slide decks will get your there in minutes.
The first is from Jim Webber entitled "HATEOAS: The Confusing Bit from REST". There is a video of Jim presenting these slides as well as a white paper about it (note: the white paper is less correct than the slides when it comes to things like specific MIME types). He walks you through a simple but illustrative (HTTP) RESTful implementation of ordering coffee. If the user agent knows the MIME type "applications/vnd.restbucks+xml" (and the associated Link rel types), then it can follow the embedded links to transition from state-to-state. And if you don't know how to do the the right thing (tm) with this MIME type, you should stop what you're doing.
It seems like the Twitter API is moving away from HATEOAS. Brian Mulloy has a nice blog post about this (from which I took the image at the top of this post). The picture nicely summarizes that from an HTML representation of a Tweet there are all manner of state transitions available, but the equivalent json representation is effectively a dead-end; the possible state transitions have to be in the mind of the programmer and encoded in the application. Their API returns MIME types of "application/json" just like 1000 other APIs and it is up to your program to sort out the details. Twitter's 1.1 API, with things like removing support for RSS, is designed for lock-in and not abstract ideals like HATEOAS. Arguably all the json APIs, with their ad-hoc methods for including links and uniform MIME type, are a step away from HATEOAS (see the stackoverflow.com discussion).
The second presentation also address a pet peeve of mine: API deprecation (e.g., my infatuation with Topsy has tempered after they crashed all the links I had created -- grrr.). The presentation "WHY HATEOAS: A simple case study on the often ignored REST constraint" from Wayne Lee walks through a proper way to define your API with an eye to new versions, feature evolution, etc.
Again, I'm sure there are many other quality REST and HATEOAS resources but I wanted to gather the couple that I had found useful into one place and not just have them buried in class notes. Apologies for being about five years late to the party.