Hi Adrian,
This is probably one of my favorite improvisations
http://www.youtube.com/watch?v=YiMnLsJK-qk and I happen to think the guy is pretty good at it as well (which is pretty evident). I've listened to this piece enough times (and liked it enough) that I've been able to come up with my own improvisation of this improvisation that sounds ok to me. I don't know, but when improvising you're going into uncharted territory and you have to be able to "wing it" as I call it. It's not like someone is going to say "you did a terrible job at improvising that song" unless you really did butcher things up which to me is pretty hard to do unless you have no ear or feel for things at all.
I'm less than an ok-ish organist for sure
, but to me it's like playing any other instrument and coming up with a good riff (like the rock guitarist that pulls off the good sounding crazy lead), it's all about what sounds good to you and how you take a familiar (or even made up song) and add to it or change it in how you like, maybe you slow the tempo down somewhere along the way and add someting extra, maybe you go the other way and do the opposite, to me there's really no rules as long as in the end it flows.
If you are able to play a song completely by memory and need no music in front of you, to me that's a good start to improvise it. However, I've met some that have to stick strictly to the sheet music in front of them at all times and can't even play a basic version of "happy birthday" without it. If you can play from memory, know the progression of the song you want to improvise, know the chords and so on, it's all about how you alter and piece things together, keep a smooth flow going, make it sound legitimate and so on.
Where does one start? Are you able to listen to a song, get it in your head, and play at least a rendition of it by ear? Do you ever sit down at the console (with no sheet music) and just make things up as you go or goof around and play around with songs / familiar tunes and just make music? I take a few minutes to do this by habit each time I sit down. If you're able to do this you're well on your way to improvising.
Marc