Use the thinnest white paper you can find. The white paper in Origami kits is perfect; so is really cheap typing paper.
Fold it as many times as possible -- in half, in half again, into a triangle, into a small triangle and - if you can manage it - into an even smaller triangle.
Make lots of small cuts instead of a few big ones, and make the cuts as deep as you can without actually going all the way through. The more paper you cut out, the lacier the snowflake will be. (Nail scissors are really handy for this.)
And, the most important step: place the finished (unfolded) paper snowflake between two sheets of wax paper -- the old-fashioned kind, not the kind made for microwaving -- and iron the wax paper thoroughly on both sides. Let them cool and peel off the wax paper. This makes the flakes perfectly flat and gives them just the right amount of weight.
If you're hanging them on a tree, use dark green thread and they'll seem to float on the branches.