The idea behind a beacon is that its a small device which can send a signal directly to someone's smart phone if they get close enough to it. So for example, if you're in the lawn mower section of Homebase then 'ping' you get a message on your phone that the company passes onto you via Facebook which says 'hey, we've got 10% off today!'.
There are limitations of course - you have to have a smartphone in the first place, it needs to know where you are, and you would have to have notifications turned on. If all of those criteria are in place, then you might soon be discovering this new form of advertising. However, at the moment, it's only slowly being rolled out, and in the United States only, so if you hate the idea you don't need to worry about it yet.
It's got a lot of other potential uses though, especially within a library setting. Student is in a particular stack and 'ping' gets a message telling her that there's also useful information on the third floor, or that there is an exhibition on about the subject in the local museum or a link to the library website for example. When it's time to close for an evening, 'ping' you get told that by your smartphone. I'm not sure how personalised this is yet, or is going to be, but if you can focus on particular individuals they could get a message when they come into the library that they need to return overdue books, or that the item they requested is available, or the answer to the question they asked yesterday is 'xyz'. I can see lots of useful ways in which this could work.
Of course, there's also the question about how intrusive this kind of approach would be. Well, you'd be able to turn it off of course in your settings. There are also privacy questions, so I'm guessing that you'd want people to be able to opt into a system rather than forcing it on them.
This is all pie in the sky at the moment - there's no suggestion (yet) that Facebook is going to be rolling this out beyond the business and retail sector, but if it proves successful and popular there, I think libraries would be an obvious next step.
If you're interested there's more directly on Facebook's own site.