Of course Jonathan Swift's "A Modest Proposal" is not intended to be taken seriously, not by those who originally read it and not by modern readers. Swift's use of irony serves to shock his readers in the hope that something will be done to remedy the problems he is highlighting.
Understatement is also known as verbal irony; it is a contrast between what the speaker says and what he means. The greatest example of verbal irony is the title, for after the reader has read a few paragraphs of the pamphlet, he knows that Swift's proposal is anything but modest--it is outrageous, in fact. Note the following:
I shall now therefore humbly propose my own thoughts, which I hope will not be lyable to the least Objection.
Note his downplaying (understating) his ideas when he says humbly and "not [liable] to the least objection." With this set-up, readers are just not expecting to hear him say we should sell and eat children.
Overstatement is also a kind of verbal irony Swift uses in this writing. He treats this shockingly inhumane idea as a scientific plan. His scholarly, overly formal style can be seen here:
As to my own part, having turned my thoughts for many Years, upon this important Subject, and maturely weighed the several Schemes of other Projectors, I have always found them grossly mistaken in their computation.
This presentation of a horrifyingly cruel plan as a purely scientific calculation is part of Swift's irony.
Swift also uses details (specifics) to add to the irony of the work. Note how many statistics he uses to make his point (even though they are blatantly untrue) and how many important (but fictional) people he cites to make his case. The speaker claims to have heard this plan from someone else, but his friend
confessed, that this expedient was put into his head by the famous Sallmanaazor, a Native of the Island Formosa, who came from thence to London, above twenty Years ago, and in Conversation told my friend, that in his Country
Such a plan worked beautifully. It is all nonsense, but it sounds so official because of the specific details Swift has included.
"A Modest Proposal" is one of the most recognizable and referenced satires in literature because it uses irony so effectively.