For a watch to be “triple signed” means that the manufacturer name (or trademark) is stamped on the case (most often the inside or outside of the case back), movement and dial. Depending on the watch, you can also find names and trademarks on other parts, such as the crystal or crown, and on the strap and/or buckle, or bracelet. If you are looking for originality, you’ll want to know which you should find for a particular timepiece. But as with anything, that doesn’t guarantee that those components came with that specific watch when it was first sold – just that they are correct for that reference.
There are watches that can be original, and not be triple signed, which happened in such situations as a watch being cased in a different country than the one in which produced.