@SMickey,
There was a social-political movement, called the Beat Generation which occurred right after WW II, just before the hippy movement (pre-1965), called
beatniks. These non-conformist, pacifist groups of people would collect at coffee houses and recite poetry, accompanied with bongo drums, to their dark, nihilistic poetry. That poetry was called 'beat' (bongos punctuated the recitation).
Leading beat poets who were popular were Ferlinghetti and Alan Ginsburg. Kerouac, author, was quite popular then and wrote hugely popular best-sellers
On The Road and
Dharma Bums.
"Beat poetry evolved during the 1940s in both New York City and on the west coast, although San Francisco became the heart of the movement in the early 1950s. The end of World War II left poets like Allen Ginsberg, Gary Snyder, Lawrence Ferlinghetti and Gregory Corso questioning mainstream politics and culture."
So, beat poetry night might be a night that a group of poets or would-be poets get together and socialize, reciting free-form poems to each others..sort of like amateur night.