Salesforce just released the REST version of their API, and while there's a Java example, there's no sample code for other languages. Since I'll be calling it from PHP, I used their documentation to build my own sample code. The source is available at http://github.com/petewarden/salesforce_restphp_example/ and you can see a live version running at https://www.openheatmap.com/labs/salesforce_restphp_example/. The code demonstrates how to authenticate, get an access token and then call the API to grab information about the sales accounts for the current user.
To use the API at all, you'll need a server setup up with an SSL certificate and https, since the OAuth 2.0 authentication requires a secure connection. I found this guide from Ubuntu useful in getting that set up, and bought my certificate from GoDaddy.
With that sorted out, go to http://developer.force.com/join to create a Developer Edition salesforce account. You'll also want to sign up for the REST API preview beta program (though they're currently experiencing a few technical hiccups with the process).
Next, navigate to the Setup link in the top-right corner of the page, then click on Develop, then Remote Access. Pick an application name, and add the location where you'll be uploading the example index.php file as the callback URL. I also checked the No user approval required box even though I'm not sure exactly what it does! After you've saved you should see a screen giving you your access credentials. Copy the Consumer Key, Consumer Secret and callback URL values into the start of your copy of the index.php sample code and then upload it to the server.
Now point your web browser at the address you uploaded the sample code to. The first time through it should redirect you to a login page on the Salesforce site, ask you whether you want to let the application access your data, and then send you back to the sample code location. If all goes well, you should see a list of your sales accounts: