When playing an organ it is fairly easy to play something without having a fixed rhythm to keep to. In fact, I would even go as far as to say that an organist seldom keeps to a fixed 'beat'. It's more a case of 'play as the mood takes you' as an organist has an upper manual, a lower manual and pedals to create sounds that 'flow' to the music.
Here is a YouTube clip of an orchestra playing a piece with no 'fixed' beat. It just flows as the song progresses. Something an organist could happily try his/her hand at.
MANTOVANI - THE SONG FROM "MOULIN ROUGE" (WHERE IS YOUR HEART)
But what about playing something like this on a keyboard? Usually with a keyboard one finds a suitable Style for the tune we want to play from the many available, and then plays the melody to fit the backing.
But on most keyboards there is an option not to play with a fixed 'beat'. This is called "Free Play". Watch this video which explains clearly how you find a 'free play' configuration.
David's Freeplay style settings on Yamaha Tyros 5