Jesus appointed 12 apostles to continue His Church here on earth. Those apostles ordained new apostles, who ordained new apostles, and so on until we are here today with thousands of bishops all over the world. Since bishops cannot be at every parish all the time, we have priests to serve God and the people in a more direct way.
When discerning the Priesthood, a man has three different options. One, perhaps the most common, is to be a diocesan priest. In this case, he applies to be a seminarian under a specific diocese, and goes through formation under the bishop of the diocese. Usually this takes 6-9 years. A diocesan seminarian studies philosophy as an undergraduate, and then gets a Master of Divinity and sometimes several other degrees in theology.
Because there are so many Catholics worldwide belonging to one universal Church, communities are divided based on region and culture into areas called dioceses. The head of a diocese is a bishop. Each diocese also falls under an archdiocese, which comprises 2 or more dioceses and is led by an archbishop. An archbishop is not higher than other bishops by ordination; he is just in charge administratively of several dioceses. The Pope is also the bishop of Rome.
In the United States, there are 32 Archdioceses and 195 dioceses. Around the world there are close to 3000 dioceses. The diocese of Austin belongs to the Archdiocese of Galveston-Houston. We have 125 parishes to serve about 450,000 Catholics (and growing!). There are 200 priests and 180 deacons. You do the math...we need more priests!
The second option is to be a monastic priest. An example would be a Trappist monk. These priests enter a religious order and take vows, one of which is a vow of stability. This ensures that the priest will stay in the community for life, spending much time in prayer and work inside the community. Some monasteries have a vow of silence, so that idle talk is eliminated from daily activities where it is not needed. Usually the monks will come together several times a day to chant the Liturgy of the Hours and the Psalms. Look under 'Consecrated and Religious' for more information.
A third option for the Priesthood is to be a religious apostolic priest. An example of this kind would be a Missionary Oblate of Mary Immaculate. Apostolic religious priests are different from monastic priests in that they go out into the world to serve the people, and spend less time in their communities doing work. They are also different from diocesan priests in that they do not typically have a specific parish to run, but move around doing God's work. Many communities have a special charism, or type of work that they do. Look under 'Consecrated and Religious' for more information.