From what I understand, the devil is not a character who appears obviously in any part of the Bible.
If this is true, when was it that Satan/the devil began to terrify people who called themselves Christians, and why?
If you mean the idea of Satan (aka "the adversary"), it is mentioned in the Bible, as others have referenced in comments.
The particular image of a creepy guy with hooves, arrow-pointed tail, horns, red skin, and possibly a pointy beard, on the other hand, originated (to my knowledge) in the literature of Dante Alighieri -- specifically his Divina Commedia, or Divine Comedy, in which he narrates a journey between Hell, Purgatory, and Heaven.
While Dante concerns the work largely with medieval (it was written in the 14th century) Christian philosophy and beliefs, he drew from many other cultures in bringing the story alive. The Bible and other pre-Divine-Comedy depictions don't go into enough detail about "the adversary" to create the concrete images Dante needed for his tale.
Though the common person at the time may not have realized it, much of the symbols coursing around 14th century Italian collective consciousness were derived from the earlier Greek and Roman polytheistic cultures. Horns and hooves, for example, remained symbols of lust (especially homosexual lust), gluttony, and general excess of a morally-questionable (from the Christian perspective) variety, and so on thanks to the god Pan and his companions the Satyrs.
Dante drew on these and other symbols of older cultures in order to bring his story to life. He did such a good job of it that his description of Satan remains strong in the collective consciousness of the Christian community despite (or perhaps due to) the Bible giving so much less detail.
The devil, Satan, Lucifer, is indeed mentioned in the bible. To use @Nathan Wheeler's comment, as it has some excellent verses:
The devil is first mentioned in Genesis, the tempter of a serpent who goaded Eve into the first sin that caused all of mankind to forever fall, and is repeatedly referred to throughout the bible. However he is never discussed as a being anyone should be terrified of...according to the account on Job, Satan had to ask God for permission for every heinous action he took against Job and his family, not an entity that could wreak physical havoc at will. In 1 Peter 5:8 Satan is described as a roaring lion looking for someone to devour, that that we should (and most certainly CAN, with God's help!) resist him. A roaring lion, not a lion ripping and shredding the hapless victims of earth...just a roaring, chained lion, one who must await Gods word before he can do anything more than look and sound scary. I think one of the most important references to Satan is his temptation of Jesus as described in the Gospels. A more apt example of the true righteousness, purity and holiness of Jesus could not have been demonstrated without Jesus experiencing the same kinds of temptation every other human does at Satan's hand. So yes, I do indeed believe Satan appears quite obviously in the bible.
So long as we have faith in God, there is nothing to be afraid of, and in that context, I don't think most Christians are afraid of the devil, and certainly not terrified of him. On the contrary, I think, rather sadly, that the devil often becomes the scapegoat for the little (and sometimes big) sins in a Christians life. Thats not fear, thats a rather sly use of the devil to offload ones own guilt and sin, rather than taking personal responsibility and ultimate repentance for it.
Regarding individuals who do have a fear of Satan, I think that is more a fear of eternal death than of Satan himself. With all of our modern cinematography and the rather colorful and rich "history" of the devil as described by @HedgeMage, I think Satan becomes the scary slavemaster of hell (rather than, as Revelation tells, an ultimate eternal habitant of it himself.) I think it is eternal death in hell that people truly fear, though, and Satan is just a visualized part of that.