In an attempt to provide a detailed canonical answer I will explain some basics (much of which I'm sure you will know already and some of which I have made an educated guess about).
XFN is a simple microformat for representing relationships. It allows you to claim multiple websites as yours and identify the websites of your friends, family, contacts and colleagues etc. Many popular social networking related websites support XFN in some form (twitter, flickr, facebook, Google profiles etc.). It is commonly used in blogrolls.
My guess is that Google's Social Graph API takes advantage of data gathered by Google's Googlebot. For the Social Graph API the data derived from the Googlebot performs the function of an XFN spider. An XFN spider will traverse links building up a social graph network of links as it goes.
Several of the Social Graph API commands (lookup, otherme) run queries against data gathered by the Googlebot (so these are not a real-time query). If your site shows up in Google's searches then your site has been visited by the Googlebot and all being well there should be a good chance that the Social Graph API can return results on your site. Using the example sites kevinmarks.com, bradfitz.com and another example of tantek.com (all sites containing XFN mark-up) I was able to get API queries to return data.
The Social Graph API geturl command is performing an online version of the Python code hosted at google-sgnodemapper. This is used to create canonical form URLs for the cases where minor differences in the "me" URLs from the same site actually map to the same individual person (a single node on the social graph).
The Social Graph API testparse command runs a live real-time parse against a source page (rather than utilising the Googlebot/XFN spider data). I tried the Greasemonkey script and could not get it to work. I also tried:
<form method="post" action="https://socialgraph.googleapis.com/testparse">
<input type="hidden" name="url" value="http://kevinmarks.com/" />
<textarea name="body">
.....page source here......
</textarea>
<input type="hidden" name="contentType" value="text/html" />
<input type="submit" />
</form>
For this submission I received a "Service Unavailable. Please try again later. Error 503" response. However as this is a labs product it is likely to be unavailable from time to time.
On the Social Graph API website there is an option to force Googlebot to re-crawl your site using the Recrawl Tool. However to execute this it appears you need to be signed up to Google Engage for Agencies (which is related to AdWords).
To verify that your page contains appropriate XFN markup you can use a bookmarklet like the rel-lint XFN and rel-tag validator/lint tool or perhaps a browser plug-ins such as Chrome's META inspector v1.8.
In addition to supporting XFN, the Social Graph API also supports Friend of a Friend (FOAF). FOAF is also a format that can also be used for representing relationships using RDF. FOAF is often expressed in XML but other notations are possible (such as Notation3).
I have seen a question on the Social Graph API group asking if the API works. So it would seem that other people have seen occasional issues with using it.
The API queries themselves return JSON formatted data which you can collect and utilise as appropriate. The social graph example applications give you an online demonstration of the sort of JSON responses you can expect from passing various parameters to the API's lookup command.
I hope this helps,
