After installation of the Module with correct API key and.. I get the following error message

This message is essentially saying that there is a configuration problem with your flickr module.
More specifically it is saying that you were supposed to be sending a flickr used_id with a request for say a 'Flickr random photos' block but it was never sent which means there is no default user_id specified and no user_id was given with the 'Flickr random photos' block

I think these flickr messages that are shown to users with the 'administer flickr' role are very helpful
generally speaking although this one is probably confusing.

Perhaps if someones comes forward to write some documentation we could document these errors a little more clearly for site adminstrators

Useful link
http://www.flickr.com/services/api/flickr.photos.search.html