The reception of first clerical tonsure, and imposition of the Soutane (and Surplice), is more than a haircut. It marked, until the modern revision of the Code of Canon Law, the beginning of the clerical state in the Latin and Eastern Churches. From this point, the cleric was the Church's property so to speak; he had a gamut of obligations at law, as well as certain privileges. At that point he was incorporated, or until 1972 incardinated, into a particular order, diocese or monastery.
Of course, it's not actually a haircut (from tondere, to shear). In the symbology of the roman rite existing until the practice was abandoned after the second Vatican Council in 1972, the tonsure was just a snip or two. The tonsure has taken many forms: the roman monastic tonsure, whereby the entire head is shaved save for a corona or circle of hair; the eastern tradition, where the entire head is denuded of hair; the celtic tradition (called that of Simon Magus by its roman opponents) where a crescent is shaved at the front of the head; and that of secular clergy, where five locks of hair are cut away with a pair of barber’s scissors: from the back of the head, from the front, from each side and on the top, in the form of a cross. The latter method is also said to represent the crown of thorns.
The pileolus or zuchetto, a small skullcap seen worn by the Pope, cardinals and bishops -- but proper to any cleric or monk -- may actually have arisen simply as a small hat to cover either the tonsure or baldness, and to keep out the cold. In appearance it is similar to the Jewish yarmulke, or kippah.
Lord Brawl notes that it has been claimed that monks and clerics were tonsured so that daemons could not grab them by the hair. Regardless, along with the reception of the blessed Soutane and Surplice, there is immense significance to an otherwise bizarre haircut.
The latin word for cleric, 'clerus', actually comes from the greek 'kleros' or lot. Among the various vesting prayers taught at tonsure, is that for putting on the Soutane, or cassock:
Dominus, pars hereditatis meae et calicis mei, tu es qui restitues hereditatem meam.
O Lord, the portion of my inheritance and my chalice, You are He who will restore my inheritance. (Ps. 15)
The meaning here is that no longer does the cleric lay claim to earthly things; instead, like the Levites, it is God alone who is his lot, his portion, whilst on earth. The relationship is two-sided; the cleric is by tonsure set apart for God. Like the sacrifical lamb reserved out of the best things of the Israelites, reserved to be offered up at the Altar and to be sullied by no hand, the cleric is set apart in the same way. Profoundly mystical, this is a prefiguring of Christ, who became the perfect sacrifice.
In German this is called Aussonderung ('apart-dedication', singling-out), and refers also to the setting-apart of the bread and wine of the offertory in christian Communion liturgies. Separated from the profane, though not yet offered to God, these seemingly mundane things are made sacred, blessed for the purpose for which they are about to be used -- pre-sanctified, even (ger. Vorweihe) -- in order that they may be made fitting, or worthy to be offered up to Him. So too, ought the cleric to be viewed, and so too ought he view himself. This notion of being set apart, Aussonderung, can be linked to reasons for clerical celibacy (whilst admittedly, by the letter of the law, merely a matter of discipline). Tangentially, might the spectre of child abuse not arisen, had clergy remembered they had been specifically set apart to be holy things, of God?
Nowadays the tonsure is not normally bestowed in the Catholic Church. Indeed, in the 1983 Code of Canon Law, the clerical state begins with "the reception of the diaconate" (Canon 266§1); it is now at this point that definitive incardination occurs. However in the religious orders and other groups which use the older Pontificale Romanum, the clerical state can be said to be conferred 'early' at tonsure, as the permission to use the pre-Conciliar rites derogate particular provisions of the new Code of Canon Law (c.f. instruction Universae Ecclesiae, §§28, 35). Some may quibble, but why else use the old rites? A rite is a cultus publicum, normal acts which through ritualisation become sacralised and dedicated to a particular purpose. I say if it quacks like a goose and walks like a goose, then it is one.
Tonsure is still regularly given in the orthodox Churches both to confer the monastic state, and in non-monastic settings prior to the ordination to the order of lector. In some traditions, it also takes place during adult conversion. Like Catholicism, here it represents an offering of self to God. However, note: in the east, nuns too may be tonsured. To the best of my knowledge no Church within the Anglican Communion has ever conferred first tonsure.
In my own case I was forced to request dispensation from my promises of Incorporation. However I don't consider that that Aussonderung can be countered or set aside. Maybe the challenge of all those who no longer actively live in the clerical state -- whether in major orders or not -- is to discover just what now one has been set aside for.
Note: the meaning ascribed in rp's incardination write-up, does not cover this. Here by incardination, I mean the "canonical and perpetual enlistment in the" service of a particular diocese, order or monastery. The term is first found in the ancient roman chancery Liber Diurnus, and originates from St Gregory I (596-604). See public domain Catholic Encyclopaedia (1910).