There are several methods for giving your tatted pieces a streamlined, smooth finished look with no little square knots beginning and ending the work. One of these is hiding the ends of the threads in the work itself. You can use a fine needle to weave in the ends of the threads through the stitches already made, but if you tat tightly, this can be difficult. An alternative method is to tat over an inch or two of a folded piece of slick dental floss when you start your tatting pattern. Leave the folded loop end uncovered as well as the ends of the floss loose. When you are done with the pattern and back at the beginning of the round, cut your threads about 4 inches or so from the end of the tatting, thread those loose ends through the loop of the dental floss, and pull on the loose ends of the dental floss to work the loose ends of the tatting through the initial work. Pull the end tatting up tight with the beginning tatting and cut the loose ends off. It is a lot of work to weave in the ends of tatting, but well worth the effort for creating heirloom and museum quality pieces.
The book Impeccable Tatting by Judy Banashek is the best book I have found for clean finishing techniques.