Very well indeed, when working with its intended type of image (photographs
and suchlike).  For full-color images, the uncompressed data is normally 24
bits/pixel.  The best known lossless compression methods can compress such
data about 2:1 on average.  JPEG can typically achieve 10:1 to 20:1
compression without visible loss, bringing the effective storage requirement
down to 1 to 2 bits/pixel.  30:1 to 50:1 compression is possible with small
to moderate defects, while for very-low-quality purposes such as previews or
archive indexes, 100:1 compression is quite feasible.  An image compressed
100:1 with JPEG takes up the same space as a full-color one-tenth-scale
thumbnail image, yet it retains much more detail than such a thumbnail.

For comparison, a GIF version of the same image would start out by
sacrificing most of the color information to reduce the image to 256 colors
(8 bits/pixel).  This provides 3:1 compression.  GIF has additional "LZW"
compression built in, but LZW doesn't work very well on typical photographic
data; at most you may get 5:1 compression overall, and it's not at all
uncommon for LZW to be a net loss (i.e., less than 3:1 overall compression).
LZW *does* work well on simpler images such as line drawings, which is why
GIF handles that sort of image so well.  When a JPEG file is made from
full-color photographic data, using a quality setting just high enough to
prevent visible loss, the JPEG will typically be a factor of four or five
smaller than a GIF file made from the same data.

Gray-scale images do not compress by such large factors.  Because the human
eye is much more sensitive to brightness variations than to hue variations,
JPEG can compress hue data more heavily than brightness (gray-scale) data.
A gray-scale JPEG file is generally only about 10%-25% smaller than a
full-color JPEG file of similar visual quality.  But the uncompressed
gray-scale data is only 8 bits/pixel, or one-third the size of the color
data, so the calculated compression ratio is much lower.  The threshold of
visible loss is often around 5:1 compression for gray-scale images.

The exact threshold at which errors become visible depends on your viewing
conditions.  The smaller an individual pixel, the harder it is to see an
error; so errors are more visible on a computer screen (at 70 or so
dots/inch) than on a high-quality color printout (300 or more dots/inch).
Thus a higher-resolution image can tolerate more compression ... which is
fortunate considering it's much bigger to start with.  The compression
ratios quoted above are typical for screen viewing.  Also note that the
threshold of visible error varies considerably across images.