Definition: When making soap from scratch, you must blend the lye-water and your heated oils until the soap reaches "trace." Trace is a sort of "point of no return" in the soap making process. Once your soap "traces", the mixture will not separate back into the original oils and lye-water.
To test for trace, dip a spatula or spoon into the mix and dribble a bit of it back into the pot. If it leaves a little "trace" behind, you're there. Some people describe the "trace" as a little mound of soap that takes a second or two to disappear back into the mix. The soap does not have to be really thick just yet, it just needs to be well mixed with no streaks of remaining oil.
In the photo illustration, notice the small ridge of soap across the middle. This was the dribble off of the stirring spatula as the color was being added to this batch (the purple on the left). This is what trace looks like.
Some soap makers prefer to pour their soap into the mold at "light trace", that is, immediately after trace is reached and the soap is still very liquid. Others prefer a more "heavy trace," that is, pouring the soap after trace has occurred, and the soap has thickened considerably. The trace on the right is a moderate trace.