Beat Poetry is usually considered a San Francisco phenomenon. It also had a strong base, as most US counter culture movements do, in New York, and was greatly influenced by European poets and by non-Beat American poets such as William Carlos Williams, not to mention jazz musicians of the time.
The major point of Beat poetry is to break down pre-established notions about poetry that it should rhyme, have rhythm, be about the poet's personal life, should not be accompanied by a three-piece jazz band, should not be frank about homo-erotic experiences, etc. To a great extent it still does this. Lawrence Ferlenghetti, for instance, now publishes a weekly column in a San Francisco newspaper to prove his assertion that poetry is news.
Names to remember when looking up beat poetry include:
Lawrence Ferlinghetti, poet and owner of the very important beat bookstore/publishing house 'City Lights'.
Allen Ginsberg, author of Howl which Ferlinghetti published, after which the two of them stood trial because the FBI considered the poem obscene and tried to have it banned1.
Jack Kerouac, poet and prose author who wrote On the Road.
There are women you should know about, too. Lots of them. So many that there's a whole book about them, entitled Women of the Beat Generation.