If you are warming yourself by the fireplace, you are sitting by the hearth. The noun hearth refers to a fireplace and the area around it.
Hearth refers specifically to the paved floor of a fireplace, which may extend out into a room. Hearths are associated with home and family because the hearth was historically the main source of heat in the home, as well as where the cooking was done. As a result, family members often gathered together around the hearth. The importance of the hearth has made it symbolic of the home, as in the phrase "hearth and home."