Step 1
First, begin with a square piece of copy paper. I usually make two snowflakes for every 8.5"x11" piece of paper, so I first cut the paper in half, and then make a square from each half. This makes a snowflake about 5" across. If you're just learning to make snowflakes, using a full piece of paper for each snowflake may be easier to practice with. You can make a square from a rectangular piece of paper simply by folding one corner down to form an isosceles triangle (like the one in the next step) and trimming off the excess paper.

Step 2
Fold the square of paper diagonally to make a triangle. 

Step 3
Fold this larger triangle in half to make a smaller triangle.

Step 4
Imagine the triangle in thirds, and fold the right third over. If you want to be precise and have a protractor, each "third" in this step is a 30 degree angle.

Step 6
Cut off the top of the paper at an angle. Make sure to cut if off so that all remaining layers of paper are equal. Cutting at an angle is what makes the points of the snowflake. Eventually, you'll learn to cut at different angles to make snowflakes with points that are more or less sharp.

Step 7
This is where your imagination comes in. Begin cutting away from the sides of the paper. Usually, cutting small triangles from the sides is easiest, but don't forget to try other shapes. For this snowflake, I altered the top edge a little to make my points a different shape. I also cut out a piece with a "spiky" edge. Use your imagination! I don't use patterns for my snowflakes: I just cut as I go. My snowflakes are never the same because I don't usually plan out my snowflakes. Sometimes I find a new cut or shape that I like, and I may use the same technique on a different snowflake, but I don't copy the whole snowflake

Step 8
Unfold the paper very carefully. The snowflake will not lay flat right away, so I like to tuck them between the pages of a book for a while before displaying them. I have also ironed them (between two pieces of plain paper) to make them extra flat. Watch out though, because paper can get very hot to the touch when ironed (don't use steam!) so be sure to let it cool for a second before handling the paper, and as always excercise caution to avoid fire and burns. Ironing them also seems to make them a tiny bit stiffer, which could be good if you want to hang them rather than tape them to a surface.