This question is indeed very broad. I'm new over here but I would really want to try and give you some thoughts.
To answer briefly : It is essentially the same Sacramental act
I would begin by saying that as you can see in John 20:21-23, it is said:
Again Jesus said, “Peace be with you! As the Father has sent me, I am sending you.” [...] If you forgive anyone’s sins, their sins are forgiven; if you do not forgive them, they are not forgiven.”`
The Council of Trent (1551) declares that before the coming of Christ, penance was not a sacrament, nor is it since His coming a sacrament for those who are not baptized. The Lord then principally instituted the Sacrament of Penance, when, being raised from the dead, he breathed upon His disciples saying what is cited here-above. See Fourteenth Session - Chapter V of the Council of Trente
The Council then expresses that Christ left priests, etc. that may pronounce the sentence of forgiveness.
What is important is to note that this power does not belong to the laity. This is evident since the Papal Bull of Martin V Inter cunctas (1418).
This is more however like 'how come the Confession Sacrament is now established?' as it is and might be off-topic.
On the methods of doing it: It doesn't seem to have changed much
You should look at some articles on this old manuscript which is called the Didache. Section 14 of the Didache states:
On every Lord's Day—his special day—come together and break bread and give thanks, first confessing your sins so that your sacrifice may be pure.
Which seems to say that the first Christians did indeed confess their sins to the first priests, themselves named from the Apostles.
I would be delighted to enhance this answer if it's not precise enough.