What's he saying?
"Let me not to the marriage of true minds / Admit impediments. Love is not love"
I will not allow myself to admit that true love has any restrictions. Love is not real love
"Which alters when it alteration finds / Or bends with the remover to remove:"
If it changes in response to change, or if it allows itself to be changed by the one who is changing:
"O no! it is an ever-fixed mark / That looks on tempests and is never shaken;"
Not at all! Love is a permanent mark that persists unshaken despite the harsh winds of change;
"It is the star to every wandering bark / Whose worth's unknown, although his height be taken."
Love is the guiding, constant star for every wandering ship, a fixed point whose nature is unknown, although its height can be measured.
"Love's not Time's fool, though rosy lips and cheeks / Within his bending sickle's compass come:"
True love is not subject to the changes of Time, although beautiful faces do fall victim to the sweep of Time's curved scythe:
"Love alters not with his brief hours and weeks / But bears it out even to the edge of doom."
Love does not change with Time's hours and weeks, but endures through Time right up until the day of reckoning.
"If this be error and upon me proved / I never writ, nor no man ever loved."
If the above is false and proved against me, it would be as impossible as if I had never written anything, or if nobody had ever loved.
Why is he saying it?
Sonnet 116 is one of the most famous of the sonnets for its stalwart defense of true love. The sonnet has a relatively simple structure, with each quatrain attempting to describe what love is (or is not) and the final couplet reaffirming the poet's words by placing his own merit on the line. Note that this is one of the few sonnets in the fair lord sequence that is not addressed directly to the fair lord; the context of the sonnet, however, gives it away as an exposition of the poet's deep and enduring love for him.
The opening lines of the sonnet dive the reader into the theme at a rapid pace, accomplished in part by the use of enjambment - the continuation of a syntactic unit from one line of poetry to the next without any form of pause, e.g., "Let me not to the marriage of true minds / Admit impediments ..." This first quatrain asserts that true love is immortal and unchanging: it neither changes on its own nor allows itself to be changed, even when it encounters changes in the loved one. Quatrain two embarks on a series of seafaring metaphors to further establish the permanence of true love: in line 5 it is an "ever-fixed mark," a sea mark that navigators could use to guide their course; in line 7 it is a steadfast star (the North Star, perhaps), whose height we are able to measure (as with a quadrant) although we may know nothing of its nature (the science of stars had hardly progressed by Shakespeare's time). Both of these metaphors emphasize the constancy and dependability of true love.
Finally, quatrain three nails home the theme, with love's undying essence prevailing against the "bending sickle" of Time. Time's "hours and weeks" are "brief" compared to love's longevity, and only some great and final destruction of apocalyptic proportions could spell its doom. Note here the reference back to the nautical imagery of quatrain two with the use of the word "compass" in line 10.
Sonnet 116 closes with a rather hefty wager against the validity of the poet's words: he writes that if what he claims above is proven untrue, then he "never writ, nor no man ever loved."
In comparison with most other sonnets, sonnet 116 strikes readers as relatively simple. The metaphors are reasonably transparent, and the theme is quickly and plainly apparent. The overarching sentiment of true love's timeless and immutable nature is presented and developed in the first eight lines, but there is no twist at the third quatrain - rather a continuation of the theme. Even the couplet is but a simple statement like "there you have it." The simplicity is noteworthy, and perhaps it was deliberate: Shakespeare's goal may have been unaffected candor, sincerity of conviction. It should come as no wonder that the lines of sonnet 116 often are quoted as Shakespeare's authentic definition of love.
Another interesting fact is that this sonnet is found misnumbered (as 119) in all extant copies of the Quarto (early editions were printed in small books called quartos) but one. Even this fact has produced speculation about additional encoded meanings.