I've seen a few textures made for 3d models and I have no clue how they are made. For example, a character skin is a face very streched out to cover the vertex projection. How artists make textures like these? Simply by hand? Or there are some special tools for this?

Well first off, you usually want to create and export your UV map from your model into an image. Then, you can paint it in Photoshop or Painter, and then you can apply it as a texture map onto your model.

Here's a great tutorial for 3ds Max users:
http://www.3dtotal.com/ffa/tutorials/max/UVW_mapping_an_object/UVW_mapping_an_object1.asp