Open a browser and load the following URL into it, replacing the text in [square brackets] with values appropriate to your install:
https://[yourfogbugz]/api.asp?cmd=logon&email=[youremail]&password=[yourpassword]
Two important notes:
- If you're on FogBugz On Demand, you must use HTTPS, not HTTP.
- FireFox works really well for these kinds of API calls.
The API should give you an XML response with a token that looks something like this:
<response>
<token>tfhqpjantio6dav0er9lsjqpl5f2i8</token>
</response>
Copy the token value. This is how you use it to do a search for "foo":
https://[yourfogbugz]/api.asp?token=[yourtoken]&cmd=search&q=foo
See also the full XML API documentation.