The idea of hashtags is to make it easy to pull together a stream of tweets on a linked topic. So, for instance, when there was the recent business over Scientology and the attack on a councillor for making a humorous tweet about them, the associated hashtag was #StupidScientology. Anyone wanting to follow comments on this matter could search for this particular hashtag.
Initially I really couldn't see the point. Twitter doesn't use # as a special symbol - it would work just as well if everyone just put StupidScientology in their post. But in practice it was useful to avoid confusion in a short message by having a word or phrase intended as a search term that didn't necessarily fit with the rest of the text. However, more recently I have come to see that the hashtag is more than this - it can be used as a shorthand explanation of what is happening.
I've done a couple of tweets lately using the hashtag #teenlogic - the idea being that these tweets were demonstrating occasions when teenagers had come up with an argument that didn't work in real world logic, but worked to the teen mind. One of my posts was this:
What I realized is that the hashtag does more than act as a search tool here - it explains what's going on. Without it, I seem to be suggesting that Emily has taken my shoes. But the #teenlogic tag tells us this is a conversation with a teenager, putting the whole tweet into context. This is really remarkably powerful, allowing a lot to be condensed into a few letters. Go hashtag!