Meme: An element of a culture or system of behavior that may be considered to be passed from one individual to another by non-genetic means, especially imitation.
This is the definition presented by the Oxford dictionary. Basically this means that ideas and information mutate and replicate in much the way that human genes do. A relatively recent subculture of internet specific memes has arisen due to the easy and almost immediate exchange of information and ideas. Internet memes spread across cultures faster than any other "element of culture or system of behavior" ever could, which makes them both incredibly interesting and (often) fleeting.
One of the most interesting things about memes is that they often develop specific linguistic rules. Take the Doge meme for example. Readers acquainted with this meme will probably recognize that phrases like "Much wow. Very humor." sound correct, while there is something distinctly wrong with phrases like "Very interesting. So Funny." when printed in colorful comic sans over a picture of a Shiba Inu. This is because so called "doge-speak" has it's own grammatical rules, and may even be considered a dialect (although the grammar more closely imitates that of a pidgin. In this dialect, only the shortest form of a word should be used.
For example, if we shortened "interesting" to "interest" in the above ungrammatical phrase, we would get "Very interest", a grammatically correct phrase in doge-speak. Other rules include the pairing of "many" and "much" almost exclusively with noun phrases, and the restriction in length of doge-phrases to no more than two words. For a more extensive analysis of the doge meme, check out this article by Gretchen McCulloch.
Recently on the rise in internet meme culture is "Dat boi". This meme has spread so quickly in part because of it's almost call and response nature. For anyone acquainted with this meme, the phrase "dat boi" likely leads to the thought "o shit waddup" immediately after. It's a sort of irritating mind control that works in much the same way that saying "don't think about elephants" causes the listener to think about elephants. This factor has allowed the meme to spread easily to the material world, at least among certain circles. "Dat boi" is also often used in conjunction with other memes because the lack of grammatical rules allows for greater versatility of usage.
The two internet memes that I have discussed clearly fall under "elements of culture" in the meme definition, but there are in fact internet memes that would fall under "system of behavior." These include things like sending the entire script of The Bee Movie to someone, or translating the introduction of Avatar: The Last Airbender into another language and submitting it as an assignment. Not that I've ever done either of those things...
Memes are an essential part of human culture, and memes of the internet variety are simply the next step on the meme evolutionary train. Consider that even calling someone "meme-loving trash" is a meme, and meme hate is entirely unfounded.