Cyprian

Like Tertullian, Cyprian (~200 - 258 AD) lived in Carthage (modern Tunis) in North Africa.  He was not baptized as a Christian until the mid-240's AD, but was rapidly put into church leadership roles including bishop in 249 AD.  He was fairly wealthy from a career in the legal profession, and gave some of his wealth to the poor after his baptism.  He was martyred in 258 AD.

His writings deal with controversies in the church during his time in leadership.  He also wrote about the plague that ravaged Carthage during Emperor Valerian's reign, encouraging the Christian's to stay strong in their faith and to help the sick.

One controversy he was involved in was how to deal with believers who had renounced their faith and worshiped the emperor during times of persecution and then desired to return to the church later.  One camp refused to allow them back in the church, while the opposite camp accepted all of them without question.  Cyprian took a middle ground, one that the bishop of Rome agreed with.

Cyprian was also the first early church father who championed infant baptism.  He felt they should be baptized as early as possible. Under his influence, 66 bishops declared themselves in favour of infant baptism in 253 A.D. and within a generation it had become standard practice in the North African churches.

Cyprian based many of his arguments around the term "Catholic Church" meaning the central, authoritative, and traditional church organization that claimed descent from the Apostle Peter.  In his Treatise "On the Unity of the Church" Cyprian refers to Matthew 16:18 where Jesus tells Peter that "on this rock I will build my church" and concludes that the authority given to Peter rests only with the Catholic Church.  Anyone who disagrees with the church leaders is causing disunity and is not part of the true church.  As Broadbent states in The Pilgrim Church (p.34), Cyprian went as far as claiming, "He who is not in the Church of Christ is not a Christian."

The church had now begun to exclude any and all Christians who disagreed with the most powerful bishops on non-core doctrines.  (By "non-core" I mean any doctrines of faith not expressly addressed in the New Testament.)

This large-scale excommunication of Christian believers created new practical problems in the church that had never existed before.  If an excommunicated group baptized people in the name of Christ, are those converts really Christians?  If those converts later come to the Catholic Church and renounce whatever "heresies" that existed in that group, do they need to be baptized again?  Cyprian argued yes; other bishops argued no, but it was an issue that was not resolved in his lifetime.

For the first time in history, it was no longer enough to be baptized in the name of Christ;  Cyprian now taught that one needed to be baptized in the Catholic Church to be a true Christian.

Origen

Origen (c 185 - c 254 AD) was born in Alexandria to Christian parents, who named him Origenes Adamantius.  His father Leonides was martyred in 202 AD, and Eusebius relates a story that young Origen wanted to be martyred with his father but was thwarted by his mother who hid his clothes.

Origen was a gifted teacher of the Scriptures and discovered this gift and passion at a young age.  At age 18 (203 AD), the bishop of Alexandria, Demetrius, appointed him to succeed Clement as the leader of the theological school there.  He performed this role for a dozen years until persecution in 215 AD forced him to flee.
Origen traveled through a number of places in his lifetime, including Rome, but he stayed the longest at Caesarea in Palestine.  There he was invited by the bishops to preach in their churches.
Despite his teaching experience and ability, Origen was still a layman and Demetrius was not happy that a layman from his city was preaching in churches in other cities, despite the approval of the local bishops.  He ordered him back to Alexandria, so Origen obeyed and he took up his position at the school.  However, in 230 AD Origen was on his way to Greece when he visited Caesarea again.  The bishops there ordained him, and this really got Demetrius upset.  When Origen returned, he was removed from his teaching position in the school and excommunicated from the church.
Origen returned to Caesarea and opened a new theological school, continued his writings and teaching, and spent the remainder of his life in Palestine.  During another persecution in 250 AD, he was arrested and tortured and died a few years later as a result.

From Origen's teachings and life events, we see the early church changing in the following ways.

1. Infant baptism is now commonly practiced.  Origen states in his commentary on Romans that infants were baptized in keeping with apostolic tradition, and his writings do not appear to show he disagreed with that tradition.
2. Church leadership was becoming formalized and separated from the rest of the people in the church.  The idea of clergy and laity had already become entrenched.  The fact that Origen was a more gifted teacher than many bishops did not matter to Demetrius, who excommunicated him for preaching in other churches and being ordained without his approval.  Laymen could teach in a school of theology, but not in a church!
3. Bishops had absolute authority over church affairs in their own city.  Demetrius excommunicated Origen not because the Christians in Alexandria found him behaving sinfully or because his theology was heretical, but simply because Demetrius was jealous of his abilities and his influence with other bishops.  
4. Bishops could determine arbitrarily who was part of Christ's church and who was not.  Again, Origen was not excommunicated for immorality but for simply breaking the rules of church procedure.  This is a novel development, for following church rules was never a requirement for being a Christian in the New Testament.
5. Despite Demetrius' absolute authority in the church in Alexandria, he appears to have had no authority in Palestine, where those bishops openly ignored his decisions and allowed Origen to open a new school and teach in their churches for the final 20 years of his life.  Therefore, churches in each city still operated somewhat independently, each under their own bishop.

Tertullian

Tertullian (c 160 - c 225 AD) was a prolific Christian writer from Carthage, North Africa.  He was the first to use the term Trinity and explain that doctrine.  He never refers to himself with any title, so he may have simply been an educated layman in the church.

Tertullian was never made a saint in the Catholic Church because he held some controversial views.  However, he wrote against many heresies and covered a wide range of theological doctrines and church life, which help paint a picture of the church in his day.

In regards to church leadership, Tertullian mentions one bishop who has authority over the elders and deacons, similar to what Ignatius taught.  However, where Ignatius required the bishop to be present for baptism, Tertullian states that any believer can baptize with the bishop's permission.

"Of giving it [baptism], the chief priest (who is the bishop) has the right: in the next place, the presbyters and deacons, yet not without the bishop’s authority, on account of the honour of the Church, which being preserved, peace is preserved. Beside these, even laymen have the right; for what is equally received can be equally given."  (Tertullian, On Baptism, chapter XVII)

Tertullian argued that baptism should be permitted for children, but only when they were old enough to know Christ and to request it.

"The Lord does indeed say, 'Forbid them not to come unto me.'  Let them come then, while they are growing up; let them come while they are learning, while they are learning wither to come; let them become Christians when they have become able to know Christ."  
"Let them know how to ask for salvation, that you may seem at least to have given 'to him that asketh.' "  (Tertullian, On Baptism, chapter XVIII)

Broadbent suggests Tertullian is arguing against the baptism of infants, but Tertullian does not mention that per se.  He clearly is arguing for believer's baptism though.  Broadbent also mentions that Tertullian condemned the practice of baptizing the dead, but I could find no reference to that practice in Tertullian's writings on baptism.

Broadbent also quotes Tertullian regarding the freedom of religion.

"It is no part of religion to compel religion, which should be adopted freely, not by force."  (Broadbent, The Pilgrim Church, p. 33 from Tertullian, Apologetic To Scapula, chapter II.)

This freedom would be severely jeopardized in later years as the Catholic Church sought to compel orthodoxy through brute force.