Originally, the word meme meant something like, "The cultural counterpart of a gene, a meme is a self-propagating unit of cultural evolution (an idea or value or pattern of behavior) that is passed from one generation to another by nongenetic means (as by imitation)." Meme rhymes with dream.
Now the word seems to be used more to mean "an idea transmitted from person to person like a virus." You might say it's a fancy word for "fad".
Meme games are a variation on the old chain letter, except they take advantage of the linking capabilities in blogs, so you only need to link to the person who tagged you for the meme. Sparky tags me, I tag someone else, and on and on. The key element is that when you are tagged you must provide a link to your tagger in your blog.
Meme games are a good way to increase your Technorati ranking and get some more traffic to your blog, but they serve a more important purpose, which is to create new connections between bloggers. Say I tag you with a book meme (for example, provide your five favorite books); you comply, and tag some of your friends to do the same. They look at your post and link to my list and perhaps we have something in common.
I'm not at all sure how to apply meme games to blogs like mine that aren't personal diaries. I can't see myself choosing someone from the Liblogs blogroll and tagging them with a meme. It could have cool consequences - illuminate our shared ideas, increase the bond between bloggers, increase traffic to our sites - but it seems to be a game for another realm. Or maybe we just need to find a way to further evolve the meme to our realm.