[Note: per Jon Ericson's advice and an edit to the question on which this was an original answer, this answer has been moved, with minor edits, from here, due to edits made to that question to keep it from being a duplicate of this one.]
Calling on the name of Yahweh means that the Sethites began to engage in the public, communal worship of God.
Per David Reese
I originally heard of this in a sermon on Genesis 5 (not his sermon on 4:17-26) by Pastor David Reese. (The whole sermon is good, but jump to about 11 minutes in if you want to get right to the relevant part.) Pastor Reese points out (no doubt following Calvin and Vos) that the phrase is a synecdoche for the entirety of the worship of God:
This phrase...is almost certainly referring to the public, corporate word of God, in that it is usually combined...with other acts of public worship in the Bible.
He then cites the following passages. Remember in the following passage that he was travelling with Lot and a very large household (several hundred):
From there [Abraham] went on toward the hills east of Bethel and pitched his tent, with Bethel on the west and Ai on the east. There he built an altar to the Yahweh and called on the name of the Yahweh. —Genesis 12:8
Notice in the following passage the conjunction of "call on the name" and "worshippers":
Then will I purify the lips of the peoples,
that all of them may call on the name of the Lord
and serve him shoulder to shoulder.
From beyond the rivers of Cush
my worshippers, my scattered people,
will bring me offerings. —Zephaniah 3:9-10
I will lift up the cup of salvation
and call on the name of the Lord.
I will fulfill my vows to the Lord
in the presence of all his people. —Psalm 116:13-14
Many other psalms use the phrase in the same manner.
Per John Calvin
Calvin seems to be off the same opinion. If I understand him rightly, he is saying that in the third generation there were enough worshippers of the true God that public assemblies began to make sense:
Adam and Eve, with a few other of their children, were themselves true worshippers of God... We may readily conclude that Seth was an upright and faithful servant of God. And after he begat a son, like himself, and had a rightly constituted family, the face of the Church began distinctly to appear, and that worship of God was set up which might continue to posterity.
Per J.G. Vos
This is also explained by J.G. Vos, in his Genesis commentary (109):
This does not mean that faith in Jehovah, the covenant God of redemptive grace, first began in the time of Enos. It only means that formal public worship of Jehovah began at that time. The religion of faith in Jehovah began with Adam and Eve. By the third generation, about the time of the birth of Enos, regular public assemblies for the worship of Jehovah had originated.
We may venture the opinion that this "calling upon the name of the Lord" took place on the weekly Sabbath, and that it included prayer and offering of sacrifices. But beyond this we dare not speculate.
Jon Ericson and Amichai have brought up some interesting points in their answer, but I must disagree with their conclusion that men first began to call him Yahweh during this time. Parallelism is important to recognize in exegesis, particularly of Hebrew texts, and doubtless polytheism was beginning to about at this time. (I think there is an argument to be made that God did not reveal himself as Yahweh until the time of Moses, but that is beyond the scope of this answer.)
Though there is a sort of parallelism between "called his name" versus "called upon his name," the discontinuity is significant. I checked in my Hebrew Bible to make sure this was not simply a translational issue, and it is not. The verb is in a different from in the two cases (though, yes, it is the same verb), and in Enosh's case, the word name is preceded by the direct object marker (אֶת־שְׁמוֹ); whereas when Yahweh is mentioned, there is indeed a preposition before the name (בְּשֵׁם). Jon and Amichai have read too much into the apparent parallel. Pastor Reese, in his sermon on 4:17-26, deals with the significance of Enosh, and the stronger (antithetical) parallel is with Cain's son Enoch.
(My bigger objection to their position is that God is never named by men. He names himself to them. That is significant biblical-theologically but exceeds the scope of this post. Also I disagree that calling him Yahweh is a sign of distance; the name is covenantal and as, intimate even while supremely holy.)
Other Things of Note
Abel, and even Cain in an external manner, had already been a worshipper of the Lord (verses 3-4). Therefore, even if you don't agree with Calvin about Adam and Eve (if you don't, listen to more of Pastor Reese's sermons; the topic is not insignificant), this cannot simply mean that it was the first time since the fall anyone had worshipped the Lord at all.
It cannot mean that men now had to work their way up to God; that has already been debunked by the failure of the fig-leaves, the shedding of animal's blood to provide covering from shame for Adam and Eve, and the radical wickedness when the woman's firstborn turned out to be a seed of Satan and a killer of the seed of the woman rather than a killer of the seed of Satan.
This phrase is best understood to indicate the public, communal worship of God.