Yeats often used symbols in his poetry, whether traditional symbols or ones that he himself invented to convey the seemingly inexhaustible depths of his poetic imagination. In "The Wild Swans at Coole" the most obvious symbol would be that of the swans themselves. As with all of the symbols that Yeats uses, there is no one meaning here, but several. Primarily, swans are used to symbolize stability, a sense of place. The speaker has been counting the eponymous swans for nineteen years, and though the world around him may have changed, they have not.
It is the swans' unchanging nature that allows them to serve as yet another symbol, that of youth and vitality. We sense that the speaker has grown weary of life, apprehensive at the approaching onset of old age. (A familiar refrain in Yeats's poetry.) Yet the swans, "unwearied still," never seem to grow old, their very timelessness establishing a marked contrast to the tired, aging speaker. Though wearied by each passing year, the speaker will still retain a spark of life, however dim, so long as he can still count the swans each autumn. But if one day he should wake up to find that the swans have flown away, then what little strength and vitality he has left will depart with them.