What's the name of an image (TIFF) that's been traced, isolated from its background and saved with a transparent background?
As has already been pointed out, a TIFF is still a TIFF regardless of the contents. TIFF is a file format (Tagged Image File Format) as distinct from JPEG, PNG, PSD, etc.
Layered would refer to a file that has more than one layer as opposed to being 'flat' with all the elements combined. This is typically used to keep various elements separate from the background and/or each other to facilitate easy moving and editing of those elements. A TIFF can be layered.
Traced would generally mean that a bitmap image (such a photograph or drawing) has been converted to vector paths. This could be automated (Adobe Illustrator has this function, for instance) or could be achieved by somebody hand drawing the paths to reproduce the image. A traced image would generally be saved as something other than a TIFF (such as PDF or SVG) because TIFFs are more suited to bitmaps.
Now. I think what you are referring to is the process by which an object in an image is isolated from it's background and then saved with a transparent background so that it will sit cleanly over other elements in a composition. TIFF files support transparency so they are suitable for this purpose. There are various ways to achieve this, one of which would be to simply delete the unwanted areas with the eraser tool or using the 'magic wand' (or similar) to select the background and delete it. Another (and IMO somewhat more professional) way is to draw a Clipping Path around the part of the image that you want which is then used as a Mask when you save the file. Again, TIFFs support this functionality.
Finally, the most common industry term (at least in the UK and in my experience of other territories) for an image that has had this treatment is a Cut Out. I'm sure there are other terms which people will hopefully add in comments...
In photographic terms, retouchers would call that process a "Close Cropped" image, or "extracted" image or perhaps (though less often) a "masked" image, or a "floating" image ("a car floating on white or transparent").
CC, short for "Close Cropped" — is a common usage in the commercial photography world.
Professionally, I'd use the term "Extracted" or "Close Cropped".