The problem with the answers previously given here, and what the majority of English-speaking Christians believe, is that it comes from a perspective of the English translation of the Bible. The Bible isn't of English origin. The Old Testament was written in Hebrew (and called Tanach in that language), and the New Testament was written in Greek.
Lucifer isn't the devil. It certainly isn't the name of an angel, fallen or otherwise. The angels of God who are named in the Bible (Michael and Gabriel) have Hebrew names, even in the NT (again, which was written in Greek). The names of angels who war against God are also given by their Hebrew names (satan [Re 12:9] and abaddon [Re 9:11]). In fact, all the names of individuals who are of Hebrew ancestry are given by their Hebrew names in the NT, transliterated (not translated) into Greek, and from Greek into English.
The word Lucifer doesn't even actually occur in the original Bible texts because lucifer is a Latin word (it means light-bearer). The word in the OT translated Lucifer in Is 14:12 is heylel, used only once and derived from halal (1984 of Strong's Hebrew Concordance), a verb meaning to shine.
In Greek, the word for light-bearer is φωσφόρος, transliterated phosphoros, and used once in the NT,
καὶ ἔχομεν βεβαιότερον τὸν προφητικὸν λόγον ᾧ καλῶς ποιεῖτε προσέχοντες ὡς λύχνῳ φαίνοντι ἐν αὐχμηρῷ τόπῳ ἕως οὗ ἡμέρα διαυγάσῃ καὶ φωσφόρος ἀνατείλῃ ἐν ταῖς καρδίαις ὑμῶν
which we have in one English translation as,
We have also a more sure word of prophecy; whereunto ye do well that ye take heed , as unto a light that shineth in a dark place, until the day dawn , and the day star arise in your hearts…
– 2 Pe 1:19, KVJ
Lucifer, or more correctly, light-bearer, isn't a name. Like Christ (Anointed One), it's a title. A rank signifying delegated authority, which God bestowed on Adam (along with the armor corresponding to that rank) when He gave him dominion over the creatures of Earth, but not dominion over Earth itself (Ge 1:28). Through Adam's disobedience he surrendered his title (and thus his armor) to the devil. It's when the armor was taken Adam and Eve could see they were naked. (Ps 104:2; Ro 13:12)
Jesus was appointed Anointed One (Christ) after the Baptism of John (giving Jesus authority on Earth in parallel to the lucifer of that time). After His death, He stripped satan of the authority and armor he took from Adam (Luke 11:22), and the keys of death and hell, and rose from the dead. He then entered into the Holiest of All with His Blood, making restitution for Man's sins and cleansing the vessels therein and the armor. The Father then appointed Him lucifer (He 9:23, 24), giving Him dominion over Earth and the creatures of Earth (MT 28:18. Ge 1:26.)
Early Christians certainly didn't believe lucifer was the name of the devil. The Latin translation of the NT uses the Latin word for light-bearer once, for the same verse,
et habemus firmiorem propheticum sermonem cui bene facitis adtendentes quasi lucernae lucenti in caliginoso loco donec dies inlucescat et lucifer oriatur in cordibus vestris, 2 Pe 1:19
The reason why most English-speaking Christians believe the devil is lucifer today? because he knew that by such deceit, few would attempt to challenge him directly, thinking first of all he was given total authority, and that he still had it. But he's a liar, never having had complete authority to begin with, and being completely stripped of any authority he did have over the Earth and man. (Ge 1:28, MT 4:8; 28:18, He 2:14)
In Jesus, the Anointed One (Christ) and Light-Bearer, mankind has been raised higher than Adam was, and the devil is once again roaming the Earth, tempting men individually, but never able to again rule us all.