This is a complicated area - I wish it wasn't, but (a) there are lots of image formats, each with a different niche or purpose, and (b) there is no one best image compression - it depends on the data being compressed, and how the image will be used.

JPEG: A compression technique designed to compress grayscale and color images by discarding an adjustable amount of detail that is least likely to be noticed by a human viewer. The definition is crucial: It is designed for photographs, it does not do B&W at all, and tends to do more harm to images that are not like photographs. JPEG is lossy - it discards information from the image. The lost information is less likely to be noticed, but as you squeeze the file smaller, you will see more and more artifacts of compression. And JPEG is for human viewers - the effects of JPEG compression that are invisible to you can actually 'freak out' a computer program trying to analyze an image. There is a file format called JPEG or JFIF, that consists simply of one image, JPEG-compressed. But, several other file formats can hold JPEG-compressed images: TIFF and PDF in particular.

TIFF - The Swiss Army Knife of image file formats. 8 standard compressions at least, and a dozen proprietary ones. Literally thousands of valid variations. Several commonly used compressions, variously called CCITT, Fax, Group3, or Group4, compress scanned B&W documents very well. Because TIFF can also store multiple pages in a file, it is one of the two popular interchange for scanned multipage documents - PDF being the other. TIFF is very complex to read and edit, but compared to reading and editing a PDF file, it's child's play. On the flip side, TIFF stores just the images, with very limited additional information - no text.

TIFF + JPEG: This is a mess. Yes, there is a way to put JPEG-compressed images inside a TIFF file. However: JPEG was added in TIFF 6.0, and the spec was basically botched. The TIFF group pulled the first try and replaced it - but the damage was done. Two incompatible TIFF+JPEG formats exist, and several widely circulated software products write and read only the first 'bad' version. Dosadi software reads and writes the 2nd, 'good' TIFF+JPEG, and can read 'many' of the old TIFF+JPEG files.

PDF: These days, pretty much the standard for digital document publishing. In Dosadi products, B&W images in PDF are compressed with an LZ compression (related to LZW, similar to the widely used 'zip' compression). Grayscale and RGB images written to PDF with JPEG compression.