What is a favicon? It's that pencil thing up there at the top in the address bar. It can be any shape, but for LJ it happens to be a pencil. Google's is a rather dull G, and Amazon's is a rather funkier 'a' with an arrow.
Make 16 pixel square image, no transparency.
Open image in Irvanview, and save as favicon.ico, remembering to set the transparency at that point.
Do not, under any circs, use bloody Iconedit, which is a terrible, terrible program and doesn't support transparency either.