Echoes are remnants of shadowflux in a being. Lady Zillah mentions them in the seventh book, Flashback. When Sophie and Fitz had echoes, they caused them immense pain (both physically and mentally). Fitz had one in his heart, so he felt pain whenever he is feeling strong emotions in his heart or emotional center. Sophie had one in her head, and one in her hand. She feels pain when her anger and fear takes over, causing a "monster," metaphorically speaking, that makes it harder to calm her echoes. Everyone's echoes are different. They can take different amounts of time to go away for each person. The triggers can also differ, depending on the emotion being felt when the Shadowflux was created. For Sophie and Fitz, the echoes went away with Flori's song.
Tam also had an echo. While training with Lady Zillah, he allowed shadowflux to sink below his skin and leave a “gentle echo”. However, there were no emotions attached to the echo, so Tam’s echo did not hurt him.