The narrator in "The Signal-Man" is never directly described in the text but, by analysing his behaviour and conversation, we can learn much about his character.
First of all, the narrator is a curious kind of person. It is his curiosity which prompts his first meeting with the signalman, for example, and his desire to know about the signalman's troubles encourages him to return on the next night.
Secondly, the narrator is skeptical towards the supernatural. When the signalman confesses that he has seen a ghost, for example, the narrator responds with a number of possible explanations. This reaction demonstrates another of his character traits, namely that he is a rational man. The ghost, for instance, is explained as a "deception of sight," caused by a "disease of the delicate nerves" of the signalman's eye. Similarly, the narrator explains the "imaginary cry" as nothing more than a murmur caused by the telegraph wires.
Fourthly, we learn from the text that the narrator is not accustomed to being in a signal box. Whatever he does for a living, it certainly has nothing to do with transport. This is apparent from the way that the narrator describes the signal box; it is a gloomy and depressing "dungeon" which has an earthly smell and stifling atmosphere. In other words, it represents another world to the narrator.
Finally, the narrator is characterised as a judgmental person. While commenting on the signalman's conduct, for example, the narrator says that he is "remarkably exact and vigilant." But, in the next paragraph, the narrator shows himself to be judgmental when he comments on two minor flaws in the signalman's discharge of duty:
While he speaking to me, he twice broke off with a fallen colour, turned his face towards the little bell when it did NOT ring, opened the door of the hut...and looked out towards the red light.