There are three different categories of early church fathers:
Apostolic church fathers are the earliest church fathers. They were contemporaries of the apostles and were probably disciples of the apostles. Thus, they have a direct connection to the apostles from the New Testament.
Examples of apostolic church fathers include Clement of Rome, Ignatius of Antioch, and Polycarp of Smyrna.
2 Timothy 4:21 mentions Linus, who became the bishop of Rome. We do not have any surviving copies of Linus’s writings, but we do have writings from Clement of Rome, who became the bishop of Rome after Linus. Because Clement of Rome lived during such an early time and had this kind of a direct connection with the apostles, he is considered an apostolic church father.
Ignatius of Antioch was the bishop of Antioch, and he is considered an early church father because he may have known John directly. At the very least, his writings are heavily influenced by John.
Polycarp of Smyrna was the bishop of Smyrna, and one of his students who Irenaeus. Irenaeus wrote that Polycarp was instructed by the apostles, especially by John.
The ante-Nicene church fathers are those who came after the apostolic church fathers and before the Council of Nicaea in AD 325. Ante-Nicene church fathers include Augustine of Hippo and Chrysostom.
The post-Nicene church fathers are those who write after the Council of Nicaea. They include people like Eusebius, Jerome, and Ambrose.