A hashtag is a keyword prefixed with the "#" symbol. You add a hash tag to a tweet to indicate the topic of the message. For example "I made a nice cake. #cooking." People on twitter use hash tags for events, ideas, conferences, TV shows, and other subjects to make it easier to find topical tweets. By using a hashtag, others can use the search tool to find everybody talking about a particular topic. Hashtags were invented by early Twitter users because there was no way (or space) to add category information or other data in the body of a tweet.
The trending topics section of your home page will show hash tags alongside other currently popular keywords. The difference between a keyword and a hashtag is that a hashtag may not be a normal word. Also, hashtags are always one word.
Use a hash tag when you want other people who are interested in the same topic to find your tweet. There are a number of already established popular hash tags.
Some examples

• #followfriday (added to tweets on Friday recommending someone to follow)
• #yankees (tweets about the baseball team)
• #NYE (for New Years Eve)
• etc.

It is best not to overload your tweet with too many hashtags, use 1-3 at most.