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Learning about APIs is very specific to the service whose API you want to utilize. Some APIs support what is called "HTTP Basic Authentication", which uses a plain ol' username and password. Many Twitter desktop applications still use this method.
Some other services offer OAuth as their authentication scheme, which requires an API authentication key (actually two: a token and a secret). Twitter also happens to offer OAuth, but not all services do. Some other services may even have their own authentication scheme (remote keys, etc).
To get started learning about APIs, I'd suggest you visit the web service's website whose API you'd like to access, and search around for API documentation. Twitter's, for example, is located at http://apiwiki.twitter.com/ You may also be interested in looking at http://www.programmableweb.com/apis ProgrammableWeb's site - it has lots of resources to do with specific web service APIs. An API authentication key is required by some services, like those using OAuth, to identify your application (or web application) with the service provider (Twitter, for example).
This is used so that malicious applications can be turned off at the service provider's end, and may also be used for analytics. OAuth, an open-standard for token-based authentication, actually requires two authentication keys -- a token and a secret. They're both random strings that they give you, but the secret obviously should be kept…secret.
You'll need to learn some basic PHP (or another language that supports processing things, and not just displaying all of the code to the browser (as HTML does -- PHP is a language where you can specify if conditions for things, do some background processing, etc. HTML is just for designing the *layout* of the page). Most popular web service APIs have a PHP library of sorts, which will help you with integrating a web service into your site. I'd suggest taking a look at some PHP Tutorials, along with the API documentation on a site that offers an API.
Many of the results on a simple search for "php tutorials" should be helpful.
The next step after basic HTML would be Cascading Style Sheets(CSS), Javascript and XML. This moves you along the technologies of client side web development. Server side web development is where the API's are created and hosted for calling from client or server programs.
Authentication keys are necessary to secure server side programs and data from unauthorized calling. One set of API's you may want to call would be Google's another would be Amazon's affiliate program. In either case, that vendor will direct you on need for authentication.
If you want to develop your own server side programs and API's, you will establish the trust relationship and authentication methods for calling clients. The first server side language many web developers learn is PHP. The best place to learn the basics of any web programming component is http://www.w3schools.com/.All tutorials are free and easy to follow.
It is a great way to expand on your current skills. Google API's http://code.google.com/apis/gdata/auth.html Amazon Affiliate API's https://affiliate-program.amazon.Com/gp/advertising/api/detail/main.html.
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