The speaker Ulysses (the Roman name for "Odysseus") delivers a reflection on old age and mortality to other elderly veterans who may have fought with him in the Trojan wars. His reflection culminates in a rousing speech intended to convince this same audience of warriors to take hold of life again and to find one last great adventure. Consider the following passage:
Free hearts, free foreheads—you and I are old;
Old age hath yet his honour and his toil;
Death closes all: but something ere the end,
Some work of noble note, may yet be done,
Not unbecoming men that strove with Gods.
Death is inevitable in this poem, but he is suggesting here that there is a better way to spend the last bit of his life. Rather than waiting at home to die, unhappy and feeble, the former warriors should set out on a new adventure, knowing that they will not return. As a previous answer has pointed out, Tennyson has taken liberties with the story of The Odyssey, because all of Odysseus's crew was killed on their journey home at the end of the Trojan War. In this poem, some veterans are alive and living in Ulysses's kingdom.
As a veteran, one reason he finds life so unsatisfying is the contrast between his glory days and his current life. The last line of the passage quoted above recalls a time when he and his fellow veterans "strove with Gods," a reference to the scenes from the Odyssey and Illiad where these heroes did great things and fought mighty battles "far on the ringing plains of windy Troy."
The last few lines of the poem are famous for being emotional and motivating, where Ulysses tells his fellow warriors to take the leap into the unknown:
It may be that the gulfs will wash us down:
It may be we shall touch the Happy Isles,
And see the great Achilles, whom we knew.
Tho' much is taken, much abides; and tho'
We are not now that strength which in old days
Moved earth and heaven, that which we are, we are;
One equal temper of heroic hearts,
Made weak by time and fate, but strong in will
To strive, to seek, to find, and not to yield.
These lines once again reference the audience, men who knew and fought with Achilles. The reference to the Happy Isles and the dead hero suggest that Ulysses expects to meet with his afterlife on this adventure. But as the last lines declare, it is better to face these challenges in life head-on rather than yield passively to infirmity, old age, and death.