An icon file needs to have a large wardrobe. It never knows where it's going to have to appear!
You ask your desktop icons to show in a variety of sizes. Agenda icons appear in spacious daily or cramped monthly calendar views on a handheld or phone. Favicons can be displayed in a number of browsers, in bookmark lists, history lists and explorer bars; even occasionally at larger size on a desktop. No surprise then that dedicated icon editors make icon files that hold several images. Not only that. The pictures can be different sizes at different colour depths.
Luckily you only really need one version of your Favicon
You just need one 16 x 16 pixel image with 16 colours. Why? Unlike a desktop icon, your mini logo will rarely in fact appear outside a browser favorites list or address bar. where it's shown at small size. You don't really need to make it in a special icon editor.
Nor do you need many colours. There isn't room for them!
You can make a favicon icon as a single image in almost any art programme and convert it to an .ICO file with Irfanview or Graphic Converter. You may even find it works to save it as a BMP file - eg 'myicon.bmp' - and rename it manually to ICO - eg 'myicon.ico'.
As you see below, the small size can look reasonable on the rare occasions when it wanders outside the browser and puffs itself up large.
('Large' is 32 x 32 pixels. Width and height are both doubled ... 16 x 2 = 32... which results in four times the area.)
You do get a better display when each size has its own picture, but this is not strictly necessary for a favicon.
When you make a favicon icon outside one of the dedicated icon editors, just preview your picture at 32 x 32 pixels as well as at 16 x 16. Check that the small size will at least look acceptable if magnified.
WHY IS THE DOWNSIZED FAVICON CROSS EYED?
You'll notice the 'Large icon Reduced' picture is slightly askew. This is because it's shrunk down to an even number of pixels wide - the regulation 16. Since a central pixel is needed between the eyes, one side of the face must now be a pixel narrower than the other. At twice the width, this assymetry doesn't show. At tiny size, every pixel counts. Automatic processing doesn't always cope too well.
NEW. FAVICON IN GOLIVE. If you have Adobe's icon-mad GoLive CS, you're set for the fancy stuff without getting a special icon editor. GoLive allows you to import several versions of your favicon as Photoshop layers. If you don't want the extras, you can just import a plain graphic which Golive will set up as a favicon. Just make sure it's 16 x 16 pixels.
There are sites where you can make a Favicon online. These icon editors or generators make only the 16 x 16 pixel size, but will send your design by email as a usable favicon.
TIP. Have your symbol pattern worked out in advance if you're going to draw it online. Especially if your phone time still costs by the minute! Try it on graph paper. Or print out this grid to use.

Ideograms or picture symbols... 
Put favicon to work... 
Icon editors and favicon