What people on the internet tend to call "memes" usually start as snowclones. One of the earliest (and I know some came earlier, such as the "Mr. T Ate My Balls" fad) was, "All your base are belong to us." When the song and video made for it became popular enough, "All your ____ are belong to ____" became a popular joke template.
As technology has developed over the years, these snowclone memes have since come to involve images, sound, and video. In my first introduction to lolcat macros, all of the images were some permutation of, "I'm in ur [noun], [verb]ing ur [noun]."
Some internet trends have gone a little farther in the amount of creativity involved. For example, there are quite a few "literal videos" on YouTube, where people sing what is happening in a music video, to the tune of that music video. The most well-known one is for Total Eclipse of the Heart. It starts out:
(Pan the room)It has a lot of great lines in it, such as, "They shouldn't fence at night, or they're going to hurt the gymnasts. Why do they play football inside?"
Random use of candles, empty bottles, and cloth
And can you see me through this fan?
Creepy doll, a window, and what looks like a bathrobe
Then a dim-lit shot of dangling balls
So if we were to distill down the essence of what an "internet meme" is, it's variations on an idea, theme, or template. WhatPort80 (the SFW version of Encyclopedia Dramatica) has a pretty good explanation of this, so I won't go too far into it. But I would describe that page as being "worth reading."
The most popular and effective memes are the ones that have the most room for creativity. This is probably why one of the worst memes I've seen so far is "Subtitled Hitler," where people give their own subtitles to the pivotal scene of the German film "Der Untergang," in which Hitler finally realizes that he's lost the war. Some of the ones I've seen include "Hitler plans Burning Man" and "Hitler gets banned from World of Warcraft." The problem with these videos is that the first one you see might be entertaining, the second is less so, and after that, you just want to put your face in your hands and wonder why people even bother. Really, all you're doing is putting words onto a video and they don't even match the dialogue that much. Give up already and find something else to do.