Flannery is a bridge convention using a 2♦ opening bid to show a hand of minimal opening bid strength (11-15 high card points) with exactly four spades and five (or sometimes six) hearts.
This convention was introduced because of the awkwardness of describing such a holding, especially with four-card major systems.
Five-card major openers are somewhat better placed, because 1NT denies both three hearts and four spades, so opener may pass more safely but is nevertheless at risk of missing a potential optimum 4-3 Moysian fit in spades.
Flannery is also useful when playing five-card majors in conjunctions with forcing notrump, to prevent opener from having to respond in a nonsuit after 1♥ – 1NT when not strong enough to reverse.
The responses are: "Anti-Flannery" is a variation used in systems which use the canapé treatment, where a shorter suit is opened first.