Note: This tutorial is written for a right-handed tatter. Left-hander may have to make some adjustments.
First up - How to make a ring.
Pinch the thread with the thumb and forefinger. Wrap the thread around your fingers moving away from the ball in the direction of the shuttle, and bringing it up to meet the beginning. Pinch the two threads together. This loop round the fingers will be used to make the double stitches. The thread from the shuttle is referred to as the core thread.
With the shuttle, make a double stitch over the core thread, refer to the earlier tutorial on how to make a double stitch.
Make several double stitches and picots over this core thread, eg. 4-4-4-4. Remove the thread from around the fingers and you should have something like this.
To close the ring, hold the double stitches just made between the thumb and the forefinger of the left hand and with the right hand, pull the shuttle thread in a clockwise motion. A properly closed ring looks like this.
Next - How to make a chain
Reverse the position of the ring so that it now points downwards like this (left). The position of the ball thread is on the left of the ring and the shuttle thread on the right. Grip the ring with the thumb and forefinger of the left hand and wrap the thread from the ball round the fingers. Do not bring the thread back to the beginning, instead anchor it by the little finger by looping around it a few times. The thread from the ball will be flipped to make the double stitches for a chain, and the shuttle thread becomes the core thread.
Finally, a join -
Reverse work again so that now the ring is back in the original position and the chain is pointing downwards. Loop the shuttle thread round the fingers to make the next ring. After making the first four double stitches, bring the last picot of the first ring close to the finger thread of the ring in progress. Pull up this thread with a hook through the picot and pass the shuttle through it. Make sure that the thread that is pulled up through the picot is not twisted. bring the shuttle through and pull back the finger thread to close the join. Make sure that the shuttle thread is not pulled through the picot when closing the join.
Complete the rest of the ring and close it. Now, you have two rings joined together with a chain.
Click here for a printable version of the tutorial.