The fish (Latin: piscis) is a symbol of Jesus Christ, from the name and title of our blessed Lord:
Ι᾿ησοῦς Χριστός, θεοῦ Υἱός, Σωτήρ
Iesous Christos, Theou Yios, Soter
Jesus Christ, God the Son, our Savior.
The early Christians, taking the first letter of each word, formed ΙΧΘΥΣ, (ichthys) the Greek word for fish.
Clement of Alexandria, in writing of the ornaments which a Christian may consistently wear, mentions the fish as a proper device for a ring, and says that it may serve to remind the Christian of the origin of his spiritual life. (Paedagogus, III)
Christians came to be called Pisciculi, little fishes, with reference to their regeneration in the waters of baptism.
On Baptism (Tertullian)
Chapter 1. Introduction.
Happy is our sacrament of water, in that, by washing away the sins of our early blindness, we are set free and admitted into eternal life! A treatise on this matter will not be superfluous; instructing not only such as are just becoming formed (in the faith), but them who, content with having simply believed, without full examination of the grounds of the traditions, carry (in mind), through ignorance, an untried though probable faith. The consequence is, that a viper of the Cainite heresy, lately conversant in this quarter, has carried away a great number with her most venomous doctrine, making it her first aim to destroy baptism. Which is quite in accordance with nature; for vipers and asps and basilisks themselves generally do affect arid and waterless places. But we, little fishes, after the example of our ΙΧΘΥΣ Jesus Christ, are born in water, nor have we safety in any other way than by permanently abiding in water; so that most monstrous creature, who had no right to teach even sound doctrine, knew full well how to kill the little fishes, by taking them away from the water!