You can queue up to get a ticket! Here's what the BBC Proms Guide for this year says : " Standing tickets are usually still available on the Last Night itself (priced �5, one per person ). No previous ticket purchases are necessary.Just join the queue on the South Steps, east side (Arena) or the top of Bremner Road, right side (Gallery) during the afternoon and you may well be lucky"
Good luck if you try it! ( 'Gallery' means up 'in the Gods'. 'Arena' means the oval area right in front of the orchestra.)
Seats are all sold by ballot, in that even people who book for a minimum of six concerts and book in the early, priority, booking period (ended 19th of May) are not absolutely guaranteed a Last Night seat but they'd be very unlucky not to get one.There is also an open ballot, which anyone can enter in the hope of getting a ticket.
NEVER buy off a tout.Last year I saw one unhappy man refused entry to the Last Night because his ticket was a forgery.