This is the kind of trick it is best to learn by watching someone doing it, rather than by reading about it, but I will explain it here never the less.
A flourish is a move involving one club in one hand, where the club seems to wind around the fingers and thumb in a lovely flowing motion. Start with the club held upside down in your hand as shown:
Your index finger should roughly be over the join between the handle and the body of the club. There are three further positions through which the club must pass during a flourish and the best way to learn the trick is to go through them slowly until you know them, and then put this text aside and try to link it all up into a single movement. Starting from the position shown above, you should now open out your hand palm up, and turn the club using your first finger and thumb so that it points up and is held between them, swing the body of the club towards you as you do this.
Now open up a gap between your index and middle fingers and allow the club to topple forwards so that the body goes through the gap. Grip the club here body down, handle up.:
Finally turn your hand over so that the palm faces down, and point the club forward so that the handle runs along underneath your forearm::
You can either finish here, or lift the first finger over next to the middle finger and turn your wrist to get the club into throwing position. Try to keep the club turning in roughly the same plane throughout the flourish. Your forearm doesn't move much, only to get out of the way of the handle as it swings.
Often when someone goes through these movements for the first time they are sure that something is missing - when a flourish is performed well it looks a lot more complicated that it actually is.
Flourishing while Juggling
Once the flourish on its own has been mastered, it must be introduced it into a juggling pattern. It is very difficult to fit one into a regular cascade with single spins, so I recommend the following method to start with:
After having caught the club to be flourished in your right hand, make the next left-hand throw a double going straight up (rather than a single going back to the right, as you would in a normal cascade). While the double is in the air on the left-hand side, the right hand does a flourish. The double comes back down to the left hand, and then you throw a single from the left hand over to the right, and continue with the cascade.
Try this combination without the flourish to start with, i.e. from a cascade throw a left-hand double straight up to be caught back in the same hand, and when it comess down again continue the cascade as before. Do this a few times, and in particular concentrate on the club that is held in the right hand while the left double goes up and down - this is the club that you will flourish. You can also practice with two clubs, try a flourish in one hand while the other throws a double straight up, just to get your brain used to doing the two moves together.
To incorporate the flourish into a juggling pattern, you must first of all catch one club upside down in the right hand. Do this by putting less spin than usual on the left-hand throw, and by catching the club higher than normal in the right hand. Start the flourish as soon as you make the catch, don't wait until you throw the double.
I think that the above method is the easiest way to do a single flourish while juggling, however there are variations that allow you to do more than one. For example, you could keep throwing doubles up with the left hand (i.e. do two in one hand) while the right hand flourishes the same club over and over. Alternatively, keep repeating the 'single flourish' trick above, every time a club lands in the right hand. If you use the last variation you should find that it is the same club that is thrown up as a double each time.
It is also possible to flourish every club with both hands, this is known as continuous flourishes. Juggle a cascade with slow doubles, and try to fit a flourish in each time you catch a club. This pattern is slightly quicker than just flourishing with one hand as described previously, but as long as you are confident of the movements in both hands it shouldn't be too difficult.
There is a little-known of extension to the flourish which, apart from making the move last longer, also results in the club finishing in the same position as it started (i.e. upside down - see first diagram above). Instead of finishing the move in the normal way, tilt the hand to the left and get the middle, ring and little fingers out of the way - the body of the club should fall towards the thumb, pivoting around the first finger to be caught in the starting position ready to be re-flourished. The first few times you try this it may seem that the movements are unnatural and jerky, but after you have done it a hundred times or so you should have found the right wrist movements.
One of the nicest flourish variations in my opinion is 'double flourishing', where each club is flourished twice before being thrown. The time required to make a second flourish necessitates an extra 2 in 1 hand throw on the other side of the pattern.
It is also possible to flourish every club in a single-spin cascade. Although this is a very fast trick it is quite possible, as long as your flourishes are good in both hands. You must use high singles, and reach up to catch each club, just after it peaks, rather than letting it fall. The clubs don't stay in your hands long enough to be flourished in a normal cascade, so by grabbing the club early, and carrying it down instead of just letting it drop, you can buy some extra time. Catch the clubs right in the middle so that the flourish is as small as possible, and requires the minimum amount of energy. Start the flourishing action as soon as you catch the club, and continue it as the carry the club down.