Twitter hashtags are the strange little oddities that you find on lots of tweets, and they use the hash symbol # followed by something else, so an example would be #cilip. They're very useful, but can be a bit puzzling, so let's shine a light on them.
The first thing to know about hashtags is that they are entirely informal. They're not something that Twitter has instituted, and there are very few 'rules' about how they are used. Actually scratch that - there are no rules at all, just a general code of practice. You don't have to ask any permission to use them, there's no central authority, and you can use them pretty much however you'd like to.
Why do they exist? Well, suppose that you're interested in a particular subject, conference, television programme, event or other similar 'thing'. If anyone you follow tweets about it, the chances are that you'll see it, but what about all the other folk who are also talking about the same thing who you don't follow? You'll be missing all their tweets and information. Now, you can of course do a search at any number of search engines to try and find appropriate information, but how do you find it? There are plenty of different ways that you can refer to something - in short, there's no controlled vocabulary. However, if people can agree on a specific word to use in their tweets, this can in part be overcome, as you can just search for that word, and to enhance, or emphasis it, the # helps identify the keyword even more. Consequently, you can then just search for #keyword and you'll see all the tweets that people are writing, all gathered together in one place.
A second way in which tweets are used is to emphasis something - it's almost like writing in bold or italics - people don't really expect you to search for similar tweets, they're just making their own personal point, such as #gladitsfriday although sometimes these hashtags do seem to take on a life of their own, and you'll find that other people start to use them - I see #latenightlibrarian quite often for example.
You may also find that people will use a specific hashtag to gather a series of tweets together if they're running a training course, just to make it easy for everyone to see what everyone is up to.
There are of course disadvantages to this system. Since it's unofficial and people make it up as they go along, you'll find that the same event can be referred to by any number of hashtags, which kind of defeats the point really! The television show Big Brother might be referred to as #BB or #BB11 or #BigBrother or #BigBro but generally you'll find that after a short space of time a common usage will prevail. Some people write really long hashtags, which I think is illadvised, since every hashtag character reduces the amount of tweet space you have. This is a particular irritant with conferences - the conference organisers do seem to put great store on the year of the conference, but as Twitter is primarily real time it's pretty obvious that tweets about the conference are taking place now! There are exceptions obviously, but the shorter the tag, the better. Ideally a tag should make some sort of sense - a tweet with #cilip on it is probably going to be about CILIP - a real no brainer. However, as long as a group of people agree on a tag, it doesn't matter what it is, that's the delight of the system. Hashtags also break quite easily, since they only work with a basic character set, and symbols can break them, since it's only the first part of the tag that will be recognised, so #Web2.0 would be seen as #Web2. Another disadvantage is that if spammers see a hashtag trending they can add in their own spam, together with the tag, and this can be rather irritating.
There are a few helpful hashtag resources that you can use. Hashtags.org displays recent tweets, their content, and has a graph of the use of the term in the last few days. If you see a tag that you don't understand, What The Trend can help identify it for you. Twubs does something similar; it groups tags together, allows you to 'register' (entirely unofficially and unenforceably) a tag, and it displays popular tags. If you see a tag and you don't know what it means, try out Tagalus which shows you the definition for a tag, but again it's worth stressing that it's not official at all. What the Hashtag also does the same thing.
That's a quick overview - there's more detail at the Twitter FanWiki that you might want to take a look at as well, but what you've just read should be enough to get you up and started.