There can be various link relations in the
<head> element of HTML or XHTML pages, e.g.,
<link rel="search" …
/> for OpenSearch descriptions. For favicons IE originally used the old Windows .ico format. That's a kind of container for related Windows .bmp images in various sizes and with different numbers of colours. A favicon .ico should include an image with size 16×16 and up to 256 colours, and it is not required to offer other sizes. If compatibility with say IE5 is not your main problem 32×32 or 64×64, and more than 256 colours, might also work: Applications are supposed to pick the best image offered in an .ico for their needs, and scale it up or down if necessary. If an automatically scaled down image does not work for your icon the .ico format allows to include optimized smaller versions, notably 16×16.
Years ago there was no proper MIME type for these beasts, but
type="image/x-icon" was widely supported. Later Microsoft registered
type="image/vnd.microsoft.icon for this purpose, and modern browsers are supposed to know this type as far as they support the odd .ico format at all.
Web servers might be still configured to associate .ico with
image/x-icon, but that does not affect a correct
image/vnd.microsoft.icon in the
<head> of pages. Without a link relation IE simply tried to fetch a file favicon.ico from the relevant directory. For various reasons that was a bad idea, and modern browsers rely on explicit link relations instead of default locations for favicons and other purposes.
Other browsers and other platforms were not eager to support the odd Microsoft .ico format, but liked the idea of shortcut icons. A much better format is .png, but some old browsers including IE did not support it, or had issues with certain PNG-features. Somehow these differences resulted in two link relations,
rel="shortcut", for in essence the same purpose. Only
rel="link" is registered at the moment. It is possible to list more than one relation as in
rel="shortcut icon", and today that should be good enough if the favicon is an .ico.
If you seriously hate this image format try
rel="icon" type="image/png". GIF instead of PNG is also okay, of course excluding animated GIFs. Other formats are either unsuited for a favicon, e.g., JPEG and WebP, or still less widely supported, e.g., SVG, which will be an ideal solution for most kinds of icons.
So far for the theory of the favicon business. The practice on Blogger and other sites can be slightly more complex. IIRC you can't use the name favicon.ico, just pick something else, e.g.,
href="http://example.org/my-icon.ico" if this really is an .ico with MIME
type="image/vnd.microsoft.icon". Whatever Blogger does, the
type="icon/ico" recommended in
Tweak My Blogger makes no sense, and at least in theory (not tested) more than one link relation is unnecessary. My Blogger template contains the following line immediately before the
<link href='http://purl.net/xyzzy/xyzzy.ico' rel='shortcut icon' type='image/vnd.microsoft.icon' />
Sadly Google reader still shows the Blogger icon for the feed, and unsurprisingly a Google profile still shows the Blogger icon for a Blogger profile, but otherwise it works as expected. Maybe create a 57×57 PNG for I-Phones and I-Pads, and use the (unregistered) link relation
rel="apple-touch-icon", it can't get odder. I have no I‑Panything to test this.