A website favicon (favorites icon) is a small graphic (usually 16x16 pixels) associated with a particular website, and it appears in a browser's URL bar, tabs, and bookmarks. When I design logos, I always consider the favicon because:

1. If it looks good at 16x16 pixels, then it'll look good on anything.
2. A recognizable favicon enhances a website's identity.
3. A well-designed favicon will be easier to find in a browser's bookmarks and tabbed interface.

Certainly there are well-known logos that don't scale down into favicons very well. Just look at FedEx. The entire "FedEx" logotype is squished down, and it's barely recognizable. If I designed the favicon for FedEx, then I would just use a reversal of the subliminal arrow (between the "E" and "x"). Or I would create an animated favicon like DHL, but the animated favicon seems to only work in Firefox. If you want to see a great favicon, check out USPS. Their eagle logo is perfectly recognizable at 16x16 pixels. USPS started using the bald eagle head before modern internet browsers were invented, so the fact that it looks so good as a favicon is just a fluke. Anyway, if you're designing a logo right now, crank the resolution down to 16x16 pixels. If it's still recognizable, then you're on the right track.