Jonson's 80-line elegy to Shakespeare begins with a justification of his motives for praising the author. As in, "I know you think I can't stand this dude, but here's how it really is." And hey, that's an apt start to a poem written by a rival, right?
Jonson moves through the different reasons he might have for praising Shakespeare, but ultimately dismisses all of them except for the old classic: genuine admiration. He talks about how Shakespeare is a thousand times more awesome than Chaucer, Spenser, and all the other famous British authors of the past. He throws in a pretty famous jab that Shakespeare had "small Latin and less Greek", but then goes on to say that he doesn't even need those fancy classical languages because Shakespeare leaves Euripides, Sophocles, and all their cronies in the dust, anyway.
Then there's a shift and Jonson starts talking to the nation of Britain. He discusses the importance of bragging rights, and how the English should never let the world forget that Shakespeare came from Britain and that it was a Big Stinkin' Deal. This train of thought melts into a discussion of "nature." Not in the birds-and-trees sense, but in the natural, human, universal sense, a.k.a. the element of Shakespeare's writing that we're sure your English teachers have beaten in you since 9th grade.
The quintessential counterpart to nature, nurture, or "art" as it was often called in Jonson's time, then comes into play; according to Jonson, Shakespeare was both born with and worked for his poetic prowess. The poem concludes as all good memorials should, with Shakespeare being transformed into a star, watching over the stage as it mourns for the death of the dramatist.