Hollow kumihimo braid:

Instructional video on Youtube - Hollow kumihimo braid

These are the materials I used for this demonstration. I used four different colours of 1mm satin cord and the round kumihimo disk. I am using 8 lengths of cord of 1m each. This braid can also be made using a different number of cords (more or less). It is the same technique used. I then put all the ends of the cords together and tied a knot in the middle of them all to use that to attach to the kumihimo disk.

You want to set up the cords on your disk like the following picture. Use the dots on the disk to place your pairs of cords opposite each other regardless how many cords you are working with. You then start the movements on the disk to create your braid by taking your top left cord, jumping over the next cord moving towards the right on the disk, and then place that cord in the slot right before the next pair. Take the very next cord (the first one of the next pair you've just reached), jump over one cord moving to the right of the disk and placing it in the slot right before the very next pair of cords. Repeat this until you reach the top again and place the last cord in the empty slot to the left of the top pair. Now take the right cord from the top pair and do the same movements, but moving towards the left of the disk, until you reach the top again and place your last cord in the empty slot to the right of the top pair. 

Now your disk will look something like the last picture above. You now need to move all your cords back into the pairs so the disk looks like the original positioning (only the colours are placed differently, if you are using different colours in your braid). The top pair is already in the right position but you need to move the rest of the cords one slot over (towards each other) to form the original pairs again. Once all the cords are back in the original position you then repeat the steps from the beginning until you reach the length you want your final piece to be, and then finish off your final braid as desired.

Square kumihimo braid:

Instructional video on Youtube - Two colour square kumihimo braid

Setup you cords as shown in the picture below, and place you two different colours as pictured if you are using two colours. 

Start creating your braid by following the steps shown on the pictures below. Always take the two cords in the middle and bring them on the outside of the opposite cords, and repositioning the cords to fill the empty slots as you go along. Then turn your disk to repeat the steps on the section. And make sure you turn your disk the same way throughout your braid.