One of the most interesting characters in Chinese paper cutting is 囍. "喜" means a good thing. 囍means two good things come to you. In Chinese, we have a phrase "双喜临门", literally meaning two good things have come to your door. The "two" here really means is that more than one -- thus this phrase means more than one good thing happen to you and your family. It's a great wish, and usually used as Chinese wedding decorations.
This character looks pretty complex, but smart papercut artists have found different ways to simplify it, while still make it recognizable and interesting.
How to cut it out of a piece of paper? As long as you know how to fold paper and what a rectangle is, you can do it.
First, get a piece of red paper, preferably square paper. In Chinese, especially around the Chinese New Year, we use red paper, because it means good furtune. Other colors work too, but for this character, red is the best.
Second, fold it three times -- fold in half, then fold in half again, so you see 3 clear lines, like the step 2 in my drawing.
Third, fold it just like my step 3.
Fourth, draw seven rectangles on that folded paper.
Fifth, cut the rectangles out and you'll get a 囍! You can cut the middle part open as well, it's up to you. It you split it from the middle, you'll get two 喜.
Enjoy!