The term 'memetics' was coined by Richard Dawkins of Selfish Gene Theory fame.
A meme can be a joke, an invention, a moral story, just a group of words that rhyme, or any other replicating idea.
The idea replicates when it is communicated to another brain.
Memes can be regarded as self replicating entities in the same way as genes can be. They certainly follow the same mathematical rules. Some replicate very fast, others slow, some mutate fast, some slow, some efficiently, some less efficiently. They adapt (evolve, and sometimes die out completely (become extinct)).
Sometimes memes form mutualistic groups called meme-sets, which support each other, and help each other to be replicated. A Religion, for example is regarded as a meme-set ('follow these instruction and go to heaven' and 'tell others about these instructions' is a powerful combination). By the same token, science, the works of William Shakespeare, and this work itself are meme-sets too.
While memetics is intuitively attractive (and obvious) model to explain the transmission of ideas, unlike genetic theory it doesn't actually provide much predictive power. It is hard to define where a meme ends and begins, it is hard to define whether a meme has mutated (or how much), and there are no objective ways of defining the boundaries of a meme-set.
The main use of memetics is to put the field of genetics into perspective with other fields.
See