Socratic irony is when a person feigns ignorance about some matter as part of a strategy for exposing the ignorance of the person he is questioning.
This practice is known as Socratic irony because the Greek philosopher Socrates (469-399 BCE) used to entice casual acquaintances into discussion with him by claiming not to have any knowledge about some matter and inviting them to express their view. By questioning them closely and exposing the unsatisfactoriness of their answers Socrates would then reveal that they did not know what they had claimed to know.
Socrates' practice clearly involves irony since irony is the use of words with more than one level of meaning - on the surface the words have one meaning, but at a deeper level they have a different, often the opposite, meaning. Socrates' formula for initiating a discussion - e.g., 'I do not know how to define courage: I wonder if you can define it for me' - appears on the surface to be a straightforward request for knowledge which the other person has, or might have - it seems like 'I do not know the way to the station: I wonder if you can tell me how to get there'. But in fact it is a device which prepares the way for exposing the other person's lack of knowledge.
The word 'irony' comes originally from the Greek word eironeia (εἰρωνεία), which means dissimulation, i.e., concealment by pretence.