An anvil is a piece of steel, used by a blacksmith, which offered a stable ground for forming metal with the use of a hammer. The anvil was often put on a wooden scaffold to catch up the pressure caused by the strokes.
While stranded in San Francisco of 1893, Data tried to build a device which should help him to locate electromagnetic fields and gave Jack London a list with things he needed, including an anvil. London was able to provide him with an anvil and was surprised about Data's ability to carry this anvil with one hand. Realizing his mistake, the android quickly pretended that he had injured himself doing so. (TNG: "Time's Arrow")
When Data experienced his dreams in 2369, he saw his creator Noonian Soong working on an anvil, forming a raven wing. Data later drew a painting of Soong, which showed his work on the anvil. (TNG: "Birthright, Part I")
The Barkonian blacksmith Skoran also used an anvil for his work and created jewelry from radioactive metallic probes without knowing that these pieces were causing radiation sickness. (TNG: "Thine Own Self")