Definition of Unclean Spirit
The phrase τὸ πνεῦμα τὸ ἀκάθαρτον ("unclean spirit") and its various declensions occur in twenty-three verses (23) in the Textus Receptus.1 However, none of those scriptures explicitly describe what an "unclean spirit" is. Thankfully, Luke gives us more insight in his gospel.
In Luke 4:33, it is written,
And in the synagogue there was a man possessing a spirit of an unclean demon, and he cried out with a loud voice,
καὶ ἐν τῇ συναγωγῇ ἦν ἄνθρωπος ἔχων πνεῦμα δαιμονίου ἀκαθάρτου καὶ ἀνέκραξεν φωνῇ μεγάλῃ
Here, the phrase δαιμονίου ἀκαθάρτου appears to be a genitive of apposition, and therefore, πνεῦμα δαιμονίου ἀκαθάρτου would be understood as "a spirit, that is to say, an unclean demon."2
Luke then describes how the Lord Jesus Christ commands the spirit/unclean demon to come out of the man.
In Luke 4:35, it is written,
And Jesus rebuked him, saying, "Be quiet and come out of him!" And after the demon had thrown him in the midst, he came out of him, not even hurting him.
καὶ ἐπετίμησεν αὐτῷ ὁ Ἰησοῦς λέγων, Φιμώθητι καὶ ἔξελθε ἐξ αὐτοῦ καὶ ῥῖψαν αὐτὸν τὸ δαιμόνιον εἰς τὸ μέσον ἐξῆλθεν ἀπ᾽ αὐτοῦ μηδὲν βλάψαν αὐτόν
Whereas Luke formerly referred to it as "a spirit of an unclean demon," here he simply refers to it as a "demon."
spirit = demon = unclean demon
But, there's one more pertinent scripture.
In Luke 4:36, it is written,
And there was amazement upon everyone, and they were speaking to one another, saying, "What is this word, since he commands the unclean spirits with authority and power, and they come out!"
καὶ ἐγένετο θάμβος ἐπὶ πάντας καὶ συνελάλουν πρὸς ἀλλήλους λέγοντες Τίς ὁ λόγος οὗτος ὅτι ἐν ἐξουσίᾳ καὶ δυνάμει ἐπιτάσσει τοῖς ἀκαθάρτοις πνεύμασιν καὶ ἐξέρχονται
Notice how these people refer to the unclean demons (cp. Luke 4:33) as "unclean spirits."
spirit = demon = unclean demon = unclean spirit
Elsewhere, in the Book of Tobit, we find Asmodeus referred to as an "evil demon" (τὸ πονηρὸν δαιμόνιον).3 The same author also mentions a "demon or evil spirit" (δαιμόνιον ἢ πνεῦμα πονηρόν),4 again suggesting, like Luke, that a "demon" is equated to an "evil spirit." There is also the mention of "deceptive spirits" and "the doctrine of demons" in 1 Tim. 4:1.
A demon (δαιμόνιον/δαίμων) is a spirit (πνεῦμα), specifically an unclean spirit (πνεῦμα ἀκάθαρτον).
Definition of a Spirit
While the word "spirit" (Greek πνεῦμα) may refer to a part of man (cp. 1 Thes. 5:23), it is also used in reference to incorporal persons, for when the apostles were wondering if the risen Lord Jesus Christ was risen bodily, or if he was a mere ghost, the Lord Jesus Christ said,
Behold my hands and my feet, that it is I myself! Handle me and see, for a spirit does not have flesh and bones, as you see me have. (Luke 24:39)
Elsewhere, angels are referred to as "spirits" (Heb. 1:7, 1:14 cp. Psa. 104:4). However, it should be noted that nowhere are demons equated to evil or fallen angels even though, like good angels, evil angels would also be classified as "spirits."
Extra-biblical Account of Demons
Josephus not only mentioned demons5 but defines what they are. In the process of describing a root called "Baaras" (Βαάρας) used to exorcise demons, he wrote,6
Yet, after so many dangers, it is greatly desired because of one power: for, it quickly expels those called "demons" (now, these are spirits of wicked men which enter those who are alive and kill those who do not obtain help), and only if it is brought to the sick people.
ἔστι δὲ μετὰ τοσούτων κινδύνων διὰ μίαν ἰσχὺν περισπούδαστος: τὰ γὰρ καλούμενα δαιμόνια, ταῦτα δὲ πονηρῶν ἐστιν ἀνθρώπων πνεύματα τοῖς ζῶσιν εἰσδυόμενα καὶ κτείνοντα τοὺς βοηθείας μὴ τυγχάνοντας, αὕτη ταχέως ἐξελαύνει, κἂν προσενεχθῇ μόνον τοῖς νοσοῦσι.
There is also the account in 1 Enoch 15:8-10 which describes evil spirits/demons as being the disembodied spirits of the Nefilim ("giants").
8 And now, the giants, who are produced from the spirits and flesh, shall be called evil spirits upon 9 the earth, and on the earth shall be their dwelling. Evil spirits have proceeded from their bodies; because they are born from men and from the holy Watchers is their beginning and primal origin; 10 they shall be evil spirits on earth, and evil spirits shall they be called. (Tr. Richard Laurence)
8 καὶ νῦν οἱ γίγαντες οἱ γεννηθέντες ἀπὸ τῶν πνευμάτων καὶ σαρκὸς πνεύματα ἰσχυρὰ7 ἐπὶ τῆς γῆς, καὶ ἐν τῇ γῇ ἡ κατοίκησις αὐτῶν ἔσται. 9 πνεύματα πονηρὰ ἐξῆλθον ἀπὸ τοῦ σώματος αὐτῶν, διότι ἀπὸ τῶν ἀνωτέρων ἐγένοντο, καὶ ἐκ τῶν ἁγίων ἐγρηγόρων ἡ ἀρχὴ τῆς κτίσεως αὐτῶν καὶ ἀρχὴ θεμελίου· πνεύματα πονηρὰ κληθήσεται. 10 πνεύματα οὐρανοῦ, ἐν τῷ οὐρανῷ ἡ κατοίκησις αὐτῶν ἔσται καὶ τὰ πνεύματα ἐπὶ τῆς γῆς τὰ γεννηθέντα, ἐπὶ τῆς γῆς ἡ κατοίκησις αὐτῶν ἔσται.
Are unclean spirits (i.e., demons) visible and/or audible?
Demons are able to speak (Mark 1:34), even crying out with a loud voice (Luke 4:33). Since they enter humans (as well as other animals; cp. Luke 8:30-33), they probably use fleshly bodies as their medium to speak and act. As for being visible, there is no scripture which suggests they are; indeed, since they are spirits, and spirits do not have flesh and bones (Luke 24:39), we might assume they are not visible. However, even angels (who are also spirits; cp. Heb. 1:7) sometimes appeared to men. Yet, such appearances only occur on the rare occasions that angels are tasked with interacting with humans.
1 Matt. 10:1, 12:43; Mark 1:23, 1:26, 1:27; Mark 3:11, 3:30, 5:2, 5:8, 5:13, 6:7, 7:25, 9:25 (2); Luke 4:36, 6:18, 8:29, 9:42, 11:24; Acts 5:16, 8:7; Rev. 16:13, 18:2
2 Heinrich Meyer:
The genitive is a genitive of apposition or of nearer definition (Winer, p. 470 [E. T. 666–7]); and δαιμόνιον, which, according to Greek usage, is in itself applicable to either good or evil spirits, being used by Luke for the first time in this passage, is qualified by ἀκαθάρτου.
3 Tobit 3:8, 3:17
4 Tobit 6:8
5 Jewish Antiquities, 6.8.2, 6.11.2, 8.2.5
6 Wars of the Jews, 7.6.3
7 In v. 8, πνεύματα ἰσχυρὰ literally tranlsates as "violent spirits"