Discussing irony can be a bit of a minefield; Alanis Morrisette's song 'Ironic' was sneered at on its release for misusing the term, and the final episode of Futurama had fun with characters misusing the term. Within the literary sphere, it is irony when what is said has two meanings, a literal meaning and a different underlying meaning, especially when the meanings are directly opposed. A frequent misuse of the term is to cover wryly connected events, such as a Formula 1 driver being arrested for speeding. Also bear in mind the difference between this and ambiguity; in the latter, the different meanings are both allowed by the literal level of meaning, whereas irony is a tension between the two levels of meaning.
The term comes from the Greek, eiron, meaning 'a dissembler'; just as a dissembler disguises his / her motives, irony disguises one meaning in the phrasing of another. Dramatic irony performs this disguising by letting what is said have different meanings for the audience and for the characters who speak or hear it; an example can be found in Twelfth Night, where Olivia, believing Cesario's claim to be "a gentleman", mutters "I'll be sworn thou art" - as the audience is aware that Cesario is, in fact, the disguised Viola, this affirmation of what Olivia believes to be the truth is also a denial of the truth to the audience. (This, of course, would have been even more complicated for the original performances, in which the female characters were played by boys.) While dramatic irony is possible within poetry, it is verbal irony that is more likely to be found.
When you hear Charles Tomlinson read 'A Rose for Janet', the initial claim that what he offers the recipient is "only / an ink-and-paper rose" could be taken at face value, meaning that the poem is of less worth than a real rose would be, but the movement of the poem from there is to build its own importance into a rose that is "instantaneous / perennial / and perfect", and so, in a way, better than a real flower. The "only" is therefore packed with irony, as it both belittles and builds up the written rose. There is another example in Elizabeth Bartlett's 'Art Class': the teacher, fussing over the painting, claims "this is the problem with your work, you see: / it seems to have a life of its own." The poem makes clear that this is precisely what animates the painting, more so than the clichés the teacher wants to create - "trouble" is used ironically, then, as it is used to mean both what is bad about the painting and what is good about it.