Some might say this is obvious, but this all important step has the power to make or break a model. The all important thing is to have a conscious process where you ask yourself questions. Think about the following important paper properties:
Texture: What kind of texture might best compliment your model? Papers can be rough, smooth, crisp, soft, wrinkly, uniform, patterned, etc. For example, when folding my Smilodon populator, I decided to use the paper Unryu because I felt that the beautiful speckling and fiber patterns in the paper would help to give interest to the wide contours of the body as well as representing the fur, which can't be folded. However, this paper choice might be less suited for use as an insect, as the dense texture might detract from the small details of the insect. For very detailed models, more textures or patterns aren't better.
Color: The color can be an extremely important part of the model. If folding a representational model, carefully observe the subject to pick a color that is accurate to the actual subject color rather than a perceived or imagined color. For example, I originally thought that a good color for my mosquito would be grey. However, after looking carefully at mosquito references, I realized a more brown color would more accurately convey the subject. Additionally, if you are going to fold a model that does not use a realistic color, picking something to make a dramatic statement has the possibility to make your model stand out. For example, livien's pink ryujin or split color ancient dragon make for very memorable versions of very common models.
Thickness: Thicker is better as long as you can still manage to fold your model.
List of paper types and their properties.
Preparation
Paper - Making, Picking and Purchasing
I am still working my way out of a phase of folding only tissue foil and kami so I will expand this page as I learn and experiment more. For a thorough article on paper written by an expert, read Robert Lang's article on paper. Here is a recipe for making tissue-foil origami paper (sourced from the previously mentioned paper article by Dr. Lang).
What you will need:
a roll of household aluminum foil
several sheets of tissue paper (or similarly thin paper)
a can of spray adhesive
some newspaper
a roller
scissors.
You'll also need a location with no wind but where you can insure good ventilation; the fumes from the spray adhesive are pretty nasty (I use my parent's garage and work wearing a respirator, then open the door when I'm done to let it air out). Here's what works for me:
Lay down a few sheets of newspaper or other material to protect the surface you are working on from the spray adhesive. Cover an area larger than the sheet of paper you are planning to make.
Unroll a sheet of foil from your roll; lay it down over the newspaper. Make sure it is wrinkle free.
Spray the foil with a uniform layer of the adhesive. You should cover it thoroughly; there should be no shiny spots left. It's okay to overlap the edges of the foil; that's what the newspaper is for.
Take a sheet of tissue paper and "waft" it over the sprayed foil, letting it settle over the foil, starting from one edge. You only get one shot at this: as soon as the tissue touches the foil, it is stuck and removal is very difficult to do without introducing buckles and wrinkles! Let the tissue settle over the foil, using your hands or the roller to smooth it down, working in a single direction, to minimize the formation of pockets (which will become wrinkles). For me, it can help to begin with a corner of the tissue paper and continue to the opposite corner before laying down the rest of the paper.
Cut the two-layer sandwich free; turn it over and lay it, tissue-side down, on a sheet of newspaper and go over it everywhere with your roller to eliminate any air pockets (they will become wrinkles, but hopefully, they will be few and far between).
OPTIONAL: repeat steps three to five on the opposite side of the paper. This will make stronger, but thicker, paper.
Cut the material to square, and fold away!
Be patient. Making nice tissue foil can be frustrating and takes a lot of practice (I still frequently make pieces so horrible I have to throw them out). Tissue foil does have its limitations though, and I often find things like its tendency to wrinkle while folding and it reluctance to pre-creasing limiting when I am folding. However, I do not want to discredit its many advantages and sometimes it can be just the thing for advancing origamists. Right now I am experimenting with treating different types of paper with Methylcellulose, or MC. I have only experimented a little with this, so I will add to this section as I learn more. The last thing to address is where to purchase paper. I have also had little experience with this, but I have included some paper stores in the Links page.