Who invented the hashtag? A former Google developer named Chris Messina. He didn't invent it as part of his work life though, he created it as a Twitter user in 2007. 

The hashtag was based on the idea of channels used in Internet Relay Chat and in Jaiku, a Twitter competitor that was later bought by Google. But the real inspiration, according to a blog post Messina wrote in 2007, was that he wanted to have a “better eavesdropping experience on Twitter.”

In the blog post, Messina elaborated what he wanted these “channel tags” to do:

“What’s really interesting, however, i[s] how these channels can be used as tags within Twitter to open up entirely new possibilities.

Every time someone uses a channel tag to mark a status, not only do we know something specific about that status, but others can eavesdrop on the context of it and then join in the channel and contribute as well. Rather than trying to ping-pong discussion between one or more individuals with daisy-chained @replies, using a simple #reply means that people not in the @reply queue will be able to follow along, as people do with Flickr or Delicious tags. Furthermore, topics that enter into existing channels will become visible to those who have previously joined in the discussion.”

WSJ

Messina tried to interest Twitter in them, but was told, "those things are for nerds." Now, however, even tv news programs use hashtags to sort discussions.

Hashtags are now used to chronicle events from Syria to the Emmys, to the SuperBowl to the government shutdown. When Kate Middleton went into the hospital to deliver the #royalbaby, the hashtag was used more than 900,000 times, according to Twitter.

More about hashtags from a 2011 article from the New York Times.