Question: What vows does a Catholic priest take?

Secular clergy

The term secular clergy refers to deacons and priests who are not monastics or members of a religious institute. A diocesan priest is a Catholic, Anglican, or Eastern Orthodox priest who commits himself or herself to a certain geographical area and is ordained into the service of the citizens of a diocese,[1] a church administrative region. That includes serving the everyday needs of the people in parishes, but their activities are not limited to that of their parish.
Catholic Church
In the Catholic Church, the secular clergy are ordained ministers, such as deacons and priests, who do not belong to a religious institute. While regular clergy take religious vows of chastity, poverty, and obedience and follow the rule of life of the institute to which they belong, secular clergy do not take vows, and they live in the world at large (secularity) rather than at a religious institute. 
Canon law makes specific demands on clergy, whether regular or secular, quite apart from the obligations consequent to religious vows. Thus in the Latin Church, among other regulations, clerics other than permanent deacons "are obliged to observe perfect and perpetual continence for the sake of the kingdom of heaven and therefore are bound to celibacy"[2] and to carry out the Liturgy of the Hours daily.[2] They are forbidden to "assume public offices which entail a participation in the exercise of civil power."[2] Depending on which conference of bishops they belong to, deacons may also be required to recite the Divine Office daily. All clerics, once ordained, are forbidden from marrying or remarrying.
The teachings of the Roman Catholic Church and some scholars hold that a tradition of clerical continence existed in early Christianity, whereby married men who became priests were expected to abstain from sexual relations with their wives.[3][4]   The Council of Elvira, held in 306, before Constantine had legitimized Christianity, made it an explicit law that bishops and other clergy should not have sexual relations with their wives. Despite consistently upholding the doctrine of clerical celibacy, over the following centuries the Church experienced many difficulties in enforcing it, particularly in rural areas of Europe.  Finally, in the 12th century the Western Church declared that Holy Orders were not merely a prohibitive but a diriment canonical impediment to marriage, making marriage by priests invalid and not merely forbidden.[5][6]
The secular clergy, in which the hierarchy essentially resides, takes precedence over the regular clergy of equal rank. The episcopal office was the primary source of authority in the Church, and the secular clergy arose to assist the bishop. Only bishops can ordain Catholic clergy.[7]
One of the roots of the Philippine Revolution was the agitation of native secular priests for parish assignments. Priests of the powerful religious orders were given preferential treatment in these assignments and were usually Spaniards who trained in European chapters. The agitation led to the execution of the "Gomburza filibusteros".
St. Thomas Becket is a patron saint of secular clergy. St. John Vianney is patron saint of parish priests. St. Stephen is patron saint of deacons.
Preparation
Preparation for Catholic priesthood generally requires eight years of study beyond high school, usually including a college degree followed by 4 or more years of theology study at a seminary.[8]
At the time of their ordination as deacons (usually about a year before their ordination as priests) they promise respect and obedience to the diocesan bishop and his successors. They also promise to live in chastity, and according to the status of clergy (which includes a comparatively simple life). Diocesan priests do make vows, but they do not promise poverty, so they may own their own property, such as cars, and handle their own financial affairs.[9]
Liturgical responsibilities
In his apostolic letter Dies Domini, Pope John Paul II wrote: "Among the many activities of a parish, none is as vital or as community-forming as the Sunday celebration of the Lord's Day and his Eucharist".[10]
A diocesan priest spends much of his time preparing for and celebrating the Sacraments (Eucharist, Reconciliation, Baptism, Marriage, Anointing of the Sick, Confirmation). In the Dogmatic Constitution Lumen gentium, the Second Vatican Council teaches that the priest acting in persona Christi celebrates the Sacrifice of the Mass and administers the Sacraments. "Christ is also present through preaching and the guidance of the faithful, tasks to which the priest is personally called."[11]
There are many parishioners whom he visits, those who are ill, those who are dying, and those who are unable to travel outside their homes.  Sometimes, he is directly involved in the catechetical work of the parish and teaches catechism classes. He works with parish and finance councils that assist him in overseeing the welfare of the parish.[12] Diocesan priests may serve in myriad different capacities, these services include, but are not limited to, campus ministry, teaching, and chaplain work for hospitals or prisons.
Orthodox Church

In the Orthodox Church, the term "secular clergy" refers to married priests and deacons, as opposed to monastic clergy (hieromonks and hierodeacons). The secular clergy are sometimes referred to as "white clergy", black being the customary colour worn by monks.
Traditionally, parish priests are expected to be secular clergy rather than being monastics, as the support of a wife is considered necessary for a priest living "in the world".
See also

Catholic religious order
List of Catholic religious institutes
Religious institute (Catholic)
Secular institute
Vocational discernment in the Catholic Church