You can't understand what a meme is without understanding the principle of the selfish gene. We usually think of genetics as a function of organisms; that a species survives because its genetics work to make it able to survive. Dawkins turned this on its head. The genes are what are trying to survive and procreate, and the organisms are just complex, effective tools that the genes use for this purpose. Genes are what really matter, though. That's why, the minute your body can no longer procreate -- can no longer pass on genes -- it tends to fall apart. There's no longer an advantage to keeping it working after that point, because the survival of the organism is irrelevant, only the survival of the gene.
The idea of a meme was an outgrowth of this idea. Memes, as they were originally defined, were ideas that did the same thing that genes do. The canonical example of a meme is a religion, or an ideology. A meme is a long-lasting thing which struggles for survival against other memes, and the minds it will inhabit along the way are important only as a means to continue and spread the meme. That's why martyrdom works: the meme doesn't mind losing a host, as long as the process gets it into other hosts.
But these days if you say "meme" on the Internet odds are what people will think of is not a meme at all, but just a momentary fad. Nothing about the genetic model of the meme really applies to how these fads propogate or survive, nor is there really anything fundamentally different about how they spread today than how they used to spread 30 years ago -- it's mostly just faster, and can happen with less investment, but these are differences of quantity, not of kind.
What is mistakenly called a meme on the Internet today is more aptly compared to a microbial infection. It passes from person to person by contact, a person suffers it for a short while and then gets over it, and that's it. It survives by finding more hosts, but that's where the comparison to genetics ends, because it doesn't have any long-term survival. It doesn't have a consistent identity over time. It survives not through tenacity or ability but through sheer numbers.
It's always a pity when a really powerful, really meaningful word, which is the only way to express a unique, insightful concept, is devoured by the grinding engine of word-trivialization. Especially when it's co-opted to mean something we already had perfectly good words for. The language is lessened, and so is the scope of human thought. Having thought this through, I now regret my own contribution to the misuse and erosion of this sublime word.