If you’re looking for an exhaustive discussion of the concept meme, I recommend Wikipedia. But a quick definition would be as follows: a meme is a cultural idea, belief, or practice that evolves and spreads like an organism. The meme concept is fairly heavy duty, having spawned a whole science of memetics, and inspiring both avid adherents and vocal detractors.
In the popular usage of the Internet, memes are not quite so grand. They are something like fads or trends, only they spread much more rapidly and tend to burn out just as quickly. When people talk analytically about viral videos on YouTube–Mentos in Coke, Lazy Sunday, or Rick Rolling– they are likely to use the word meme.