What concept of fame or glory is presented in "The Seafarer?"

The short answer is that the poem presents two contrasting conceptions of glory or fame. One is connected with the heroic values of pre-Christian, Germanic culture, and concerns one's reputation for having committed brave, daring deeds -- vanquishing demons and other evil forces -- that kings reward and people praise. It's the social and political prestige that bold warriors receive for being perceived as valiant, strong, and physically brave.
The other is the sort of glory that transcends earthly reputation. It's the glory of God, and it is the best and longest-lasting type of glory.
To understand this in more detail, it is helpful to realize that the original Anglo Saxon poem uses at least four different words that modern scholars have translated as "glory." These words didn't all mean exactly the same thing to people living at the time.
For instance, consider lines 72-80, where the narrator speaks of the good actions that heroic men may do -- battling against "malice of fiends" and devils. People praise heroes for these deeds, and this praise leads to what Jonathan Glenn translates as "glory:"

that children of men after may praise him, and his glory hereafter live among angels always for ever, eternal life’s splendor, joy among noble ones.

But the original word used is "lof," which the Bosworth-Toller Anglo Saxon dictionary has defined as "praise, glory, song of praise, hymn." So here the poet is referencing a kind of glory or fame that comes from being made the subject of heroic tales told by others.
But that bit about "eternal life's splendor" (original word: blǽd) has also been translated as "glory," for instance by Gavin Bantock:

Then the children of men will praise him afterwards,
and his glory shall live with the angels,
and he shall abide in glory always,
sharing glory with the host.

The word " blǽd" has defined by the dictionary as "enjoyment, prosperity, abundance, success, blessedness, gift, reward, benefit, glory, honour." This word is used again in line 88, this time in the context of its shortcoming: It doesn't last forever.  As Jonathan Glenn translates it:

Glory is humbled, honor of earth grows old and withers, as does now every man over this Middle-Earth.

So while this sort of glory is good, its benefits are limited. As the narrator explains (lines 81-82, in a translation by Burton Raffel):

"The days are gone
When the kingdoms of earth flourished in glory;"

And here the original word Raffel translates as "glory" is especially telling: It's "onmedlan," which according to the dictionary has connotations of the puffed up sort of glory -- implying undertones of pomp, magnificence, arrogance, and boldness.
Nowadays, says the narrator, pursuing this sort of glory isn't even a realistic goal. In the past, great kings bestowed great prizes on heroes are gone. But society has changed, says the narrator. The age of great kings is over; the opportunities for heroes to aggrandize themselves in this way are past. The fame and prestige has faded. As noted above, the narrator returns to the word, "blǽd," in line 88, concluding that honor/glory/rewards-for-being-a-bold-hero has been humbled, diminished, or subdued.
The narrator goes on to note that you can't take your earthly prizes with you after death, and they won't help you when God judges you for your sins. But by living up to the harsh demands of Christianity, you may enter into a blessed state with "wuldres Ealdor," the father or master of glory (lines 123-124, translated by Jonathan Glenn):

thanks that he honored us, master of Glory, God of Eternity, in all our time. AMEN.

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References (in addition to the links below):
Marsden R. 2004. The Cambridge Old English Reader. Cambridge University Press. 
Mitchell B and Robbins FC. 1986. A Guide to Old English. Blackwell.
Raffel B. 1964. Poems from the Old English. University of Nebraska Press.
http://bosworth.ff.cuni.cz/

https://lightspill.com/poetry/oe/seafarer.html