According to The Law Dictionary, a perpetual curacy is 
“the office of a curate in a parish where there is no spiritual rector or vicar, but where a clerk (curate) is appointed to officiate there by the impropriator. 2 Burn. Ecc. Law, 55. The church or benefice tilled by a curate under these circumstances is also so called.”

From Family Search: Clergy of Church of England (in England), we can find a detailed definition of positions within the Church. 

“The incumbent of a parish is the person in charge of its spiritual well being, the “cure of souls”. He held the benefice with its income, mostly derived from its land, and might be a rector, receiving a tithe or ten per cent of the crops and grain, hay and timber (the great tithes) and of new born animals, wool, garden and other produce of the parish (the small tithes) or a vicar, receiving only the small tithes. The estimated value of each benefice in 1535 was set out in the Valor Ecclesiasticus, printed in six volumes by the Record Commissioners in 1810-34 [FHL only has Diocese of Llandaff on Film 1696528.1].

The incumbent may be known colloquially as a parson and live in the parsonage. Before the 17th century, curate was often another word for parson. Although a clergyman is technically ordained as a priest, the use of the general word priest to denote a minister of the Church of England (or Anglican Church) declined after the Reformation, being more often used in the Roman Catholic and Eastern churches. Anglican clergy were described as clerks in holy orders or clerks (a writing clerk was a “writer”). Until the 18th century a clerk who had been to university was, in Latin, called Magister. A non-graduate clerk was Dominus, a word often translated as Sir, but this does not mean that he was a knight.

The person who originally founded, built or endowed the church had the right as its patron to make presentation to the bishop of a suitable person to be its incumbent. This right (called the advowson) descended to the patron’s heirs and might be bought and sold like any other property. A college might thus buy the advowsons of lucrative benefices in order to provide positions for its future Fellows. The descent of the ownership of the advowson is recorded in the older county histories and in the Victoria County Histories.

The person presented, who might well be a relative of the patron, had usually already been ordained by his local bishop as a deacon or priest in order to celebrate mass and hear confession. He was supposed to be over 21 and of legitimate birth. It is said that the usual age at ordination was 23 years and six months.

Following approval by the bishop, the priest is then admitted to the benefice. Institution follows, putting him in charge of its spiritual cure, and then induction, which gives him rights to the land and income. The two acts are usually combined in a ceremony at the parish church (though institution may take place elsewhere) when the induction is symbolised by the archdeacon putting the bell rope into the hands of the newly instituted priest and by the latter tolling the bell.

Chaplains and curates were licensed by the bishop and, not having benefices, were not instituted or inducted. Curates, who may be assistant, temporary or stipendiary, assist the rector or vicar and are employed and paid by him. A perpetual curate, however, was nominated to a benefice by the lay owner or impropriator of its great or rectorial tithes. This lay patron kept the income from the benefice and paid (or granted land to) the curate. The latter needed only a licence from the bishop and was “perpetual” as he could only be removed by the withdrawal of that licence. A chapel of ease could be established in the outlying parts of a parish provided the bishop, patron and incumbent agreed. This might be convenient for the patron but the curate of such a place was paid from the income of the “mother church” and disputes frequently arose about the division of fees, tithes and the costs of repairs to the benefice house and the two churches.”

According to Wikipedia, the practice of a “perpetual curate” greatly affected the Church’s practice in the 19th Century.  

The Revd Charles Dodgson, perpetual curate of All Saints’ Church, Daresbury in Cheshire; and father of C.L.Dodgson, otherwise known as Lewis Carroll. All Saints had been created as a perpetual curacy in 1536 out of a chapel of ease of nearby Norton Priory – Public Domain – Wikipedia

“Perpetual Curate was a class of resident parish priest or incumbent curate within the United Church of England and Ireland. The name is found in common use mainly during the first half of the nineteenth century. The legal status of perpetual curate originated as an administrative anomaly in the 16th Century. Unlike ancient rectories and vicarages, perpetual curacies were supported by a cash stipend, usually maintained by an endowment fund, and had no ancient right to income from tithe or glebe.

In the nineteenth century, when large numbers of new churches and parochial units were needed in England and Wales politically and administratively it proved much more acceptable to elevate former chapelries to parish status, or create ecclesiastical districts with new churches within ancient parishes, than to divide existing vicarages and rectories. Under the legislation introduced to facilitate this, the parish priests of new parishes and districts, were legally perpetual curates.

There were two particularly notable effects of this early 19th century practice: compared to rectors and vicars of ancient parishes perpetual curates tended to be of uncertain social standing; and also be much less likely to be adequately paid.

Perpetual curates disappeared from view in 1868, after which they could legally call themselves vicars, but perpetual curacies remained in law until the distinct status of perpetual curate was abolished by the Pastoral Measure 1968.”

During the Victorian Period, Margaret Oliphant wrote The Perpetual Curate, a book which takes place in Oliphant’s fictional Carlingford. From the Victorian Web, we learn a bit about The Chronicles of Carlingford series. “Carlingford is the setting for Margaret Oliphant’s most famous series of novels, The Chronicles of Carlingford, published between 1863 and 1876. Carlingford is, according to Oliphant, “essentially a quiet place” with “no trade, no manufactures, no anything in particular” (The Perpetual Curate, 2). Instead, the “centre of life . . . round which everything circles is, in Carlingford, found in the clergy”. Accordingly, Oliphant often refers to the geography of the town in ecclesiastical terms. The socially advantaged inhabitants of Grange Lane nearly always belong to the Church of England; its parish living is occupied first by the evangelically-minded Mr Bury, secondly by Morley Proctor (The Rector), thirdly by Mr Morgan (The Perpetual Curate), and finally by Frank Wentworth (The Perpetual Curate). Carlingford’s poor live in Wharfside, where the parish church has established a missionary presence with services in the schoolhouse; its High Church adherents congregate a half-mile out of town at the chapel of ease of St Roque’s. Established to bear some of the weight of an overlarge parish, St. Roque’s is occupied by Frank Wentworth in The Perpetual Curate, and later by Reverend May in Phoebe, Junior.”