This answer will serve as a definition of a hashtag in terms of social networking, not mathematically or otherwise.
Hashtags refer to topics—(Especially on twitter.)
On Twitter: It’s this thing where you tweet something and then you use the hashtag (it can be a small phrase or a word) to describe your tweet overall or your thoughts on the subject. Then the hashtag becomes a link to that topic, so that when you press on it it links you to similar tweets with the same hashtags. If a hashtag is overly used on twitter by many users simultaneously it becomes a ‘trending topic’ on your sidebar, and therefore connects all the twitter community to some degree for a few minutes. (Provided you decide to hashtag your tweet as well, although when you click on the link in your sidebar you can still see their tweets)
Apparently many people have taken that concept and applied it to Facebook -it doesn’t work there. It simply has no use on Facebook, that’s why it tends to annoy many, because it implies you’re using them ‘for show’ or without understanding that they really are.
So basically, they act like tags. For instance, on instagram when you post a photo, you use hashtags to tag it so people can view it depending on what topics they’ve added to their profile.
In fact, the idea of ‘adding topics’ to your questions and profiles on Fluther is the same thing.
So technically speaking you are using them correctly to some extent, just not on the right social networking site.
So to answer your question simply: Hashtags are used to add a topic to whatever you are posting. (It doesn’t always have to make sense or relate to what you’re saying.)
Eg.: ‘Yesterday I went to the supermarket to buy some eggs, but upon arriving, I completely forgot why I went in the first place. #lol #whatanidiot #stupidity’ (That’s mainly a twitter thing.)
On Instagram: (Let’s say it’s a picture of a car): #camaro #red #vintage #old
So here’s what you’re doing, you’re trying to link as many people as possible to your photo by adding many subjects so it can be viewed by many.