A hashtag is a word within a message which is used to organise that message. So that it doesn't get confused with an ordinary word, it has a hash mark before it: "#", which is called "pound" in the USA.
For example, if I wrote "I really like cheese", I could add a hash before "cheese": "I really like #cheese". Then if anyone clicked on "#cheese", they would be taken to all the other posts containing the same hashtag.
Sometimes people use hashtags as a sort of joke, commenting on the post, rather than for organising their posts. For example, "My cousin has broken my computer. #idiot".
Facebook and Twitter both support hashtags.
A hashtag is used much like a tag. But tags are usually separate from the body of the post they're classifying, whereas hashtags are part of it.
Hashtags are an example of a folksonomy, a system where the classification is brought about by the collective action of its users, rather than being centrally imposed.