Where can I run FQL queries online in my web browser?
go to https://developers.facebook.com/tools/explorer and click the "FQL Query" button.
Here's a nice online form: https://developers.facebook.com/docs/reference/rest/fql.multiquery/
FQL is deprecated for versions v2.1 and higher and no longer available
Here's the Facebook's official page with developer tools. There used to be an FQL console there, but it's no longer available. You can use the tools there to test other FB API though.
Edit: For an actual FQL test console, you can take a look at this. It's not online like you wanted, but it's the next best thing that I could find when I went looking.
Where can I run FQL queries online in my web browser?
In Facebook Developers Graph Explorer Tool, in this format (using a simple example):
https://graph.facebook.com/fql?q=SELECT uid,name FROM user WHERE uid = me()
Update:
Graph Explorer now has a dedicated FQL query window. Press the FQL button and enter your query as:
SELECT uid,name FROM user WHERE uid = me()
query and multiquery -- on either page, scroll down to find the Test Console