Here are some miscellaneous technical clarifications (I hope)
JPEG:
JPEG is a compression method, not a file type, though it is common parlance to refer to a JFIF file containing JPEG-compressed content as a "jpeg file" and to give it a .jpg extension. TIFF files, for example, can also contain JPEG content, and in fact usually do, at least for the thumbnail.
There are both lossy and loss-less versions of JPEG compression, so it is technically incorrect to say that you always lose quality when you save a JPEG (regardless of what you meant by "JPEG"). But usually, unless you know what you are doing, you get a lossy JPEG compression with most apps and utilities.
The usual JPEG lossy compression is quite extreme, tossing out bit resolution (optionally, I think JPEGs can be 16-bit but usually aren't) and running a complex mathematical transformation, that I think is a form of Fourier transform with bandlimiting. The resulting image, unless you deliberately screw it up, appears just as good as the original to most human eyes (unless you do too many cumulative lossy saves).
The "imperceptable" detail that is lost in truncating the word length to 8 bits plus JPEG compressing, still serves a purpose. With 16 bits (or 12 bits, since many so-called 16-bit RAW formats are actually left-justified 12-bit formats) the extra detail can be exploited to recover details, such as burnt-out highlights or overly dark shadows, that appear to be lost but really aren't (until you toss out that detail and do JPEG compression). The same considerations apply to color corrections. Once you do the compression, you may find it impossible, for example, to color correct the mid-tones without simultaneously throwing off the highlights or shadows.
As an aside: The extra bits beyond 8 per color sample are so imperceptable that one can actually hide data in them (a form of cryptography) that is unrelated to the image. "Secret messages" can be sent this way.
TIFF
TIFF is a file format. The actual data in the file can be RAW, RGB, YMCK, HSV, or any other format. In fact, you can have multiple "directories" in a TIFF file. One might contain a full-resolution image while another may have a thumbnail. Yet another may contain EXIF data. While both JPEG/JFIF and TIFF may be either 8 or 16 (or even 1) bit, compressed or not, usually JPEG/JFIF files are used for compressed 8-bit data while TIFFs often contain uncompressed 16-bit data, which explains their larger size.
RAW
The RAW format contains data directly digitized from the image detector. It is necessary to adjust both for non-ideal effects in the detector and to correct for exposure/color balance/etc to produce the final image. As a typical example, my Canon EOS 20D in "RAW+JPEG" mode dumps the RAW file to the memory card and then adjusts the image and corrects the color, sharpens, saturates, etc., in a user-selectable way, truncates the word length, and saves the compressed result in a JPEG/JFIF file. In this way, it is often said that the .jpg file is analogous a photographic print while the RAW file is analogous to a photographic negative, because in a photographic negative, there is detail that doesn't appear in the final print.
Also, the so-called "proprietary RAW format" in the .CR2 files that my 20D writes is really a TIFF file. It contains the uncompressed RGB data, two different JPEG-compressed (one lossless) reduced-resolution "thumbnails," EXIF data, and 3 undocumented TIFF tags (the proprietary part).
The older .CRW files made by Canon digital cameras are CIFF files.
8-bit, etc.
The .jpg images typically made by cameras have 8 bits each for red, green, and blue. This should not be confused with 8-bit colormap format, where there are only 256 distinct colors, with 1 8-bit integer used to index a 256-color table. In this crude format, one can easily see artifacts from the limited color resolution. The 8-bit .jpg has 24 bits per pixel while the 8-bit colormap has only 8 bits per pixel.
The best color you can display on most (perhaps all?) monitors is this 24-bit color. (Well, actually this is a limitation of the video card, not the monitor.) So-called 32-bit color is actually 24-bits left-justified in a 32-bit word (actually it could be right-justified, or even centered, etc, as well) with the remaining bits set to 0 (unless you want to hide some information in them).
The so-called "16-bit" images in RAW files really means 16 bits per color sample times 3 colors per pixel, or 48 bits per pixel.
In reality, the .CR2 files made by my EOS 20D are only 12-bit images, and are "packed" into the 32-bit words, straddling the word boundaries to avoid wasting space. Because of this, the file gets even bigger if I use dcraw to convert it to an uncompressed 16-bit TIFF, for example.
In general, so-called n-bit images often have only m bits per color sample (m<n) for various values of n and m.
Is RAW sharper than JPEG?
Well, the JPEGs have truncated words and lossy compression, which makes them less sharp. But also the camera "enhances" the jpeg image, including optional digital sharpening in my 20D. So read into that what you will.
What is important to understand is that if you look at the JPEG and RAW images right "out of the box" the JPEG will look much nicer. Nevertheless, the RAW version has far more information content and could potentially look much better than the JPEG after processing. In my 20D, as I suspect in most digital cameras, the RAW file also contains information describing how the camera processed the image to make the JPEG.
When you download the RAW image to a computer, the most basic thing you can do to it is to just apply those adjustments to reproduce JPEG, perhaps uncompressed and without truncation, which is a waste of time and space on your hard drive, as you could just have used the JPEG. But if you need to modify the image, and particularly if you want to correct exposure, contrast, color balance, and highlight and shadow detail, you have much more room to work with if you start with the RAW. This is why I shoot in RAW+JPEG mode. Many of my photos will come out just right (so why waste time post-processing them?) while others will need some adjustments. The RAW file hogs so much space on the memory card that the extra space taken by the .jpg really doesn't make much difference.
I hope this helps your understanding of these technical details and I apologize in advance for any errors I may have made in this extremely long post.