The command finds the most recent tag that is reachable from a commit. If the tag points to the commit, then only the tag is shown. Otherwise, it suffixes the tag name with the number of additional commits on top of the tagged object and the abbreviated object name of the most recent commit.
By default (without --all or --tags) git describe only shows annotated tags. For more information about creating annotated tags see the -a and -s options to git-tag(1).
With something like git.git current tree, I get:
[torvalds@g5 git]$ git describe parent v1.0.4-14-g2414721
i.e. the current head of my "parent" branch is based on v1.0.4, but since it has a few commits on top of that, describe has added the number of additional commits ("14") and an abbreviated object name for the commit itself ("2414721") at the end.
The number of additional commits is the number of commits which would be displayed by "git log v1.0.4..parent". The hash suffix is "-g" + 7-char abbreviation for the tip commit of parent (which was 2414721b194453f058079d897d13c4e377f92dc6). The "g" prefix stands for "git" and is used to allow describing the version of a software depending on the SCM the software is managed with. This is useful in an environment where people may use different SCMs.
Doing a git describe on a tag-name will just show the tag name:
[torvalds@g5 git]$ git describe v1.0.4 v1.0.4
With --all, the command can use branch heads as references, so the output shows the reference path as well:
[torvalds@g5 git]$ git describe --all --abbrev=4 v1.0.5^2 tags/v1.0.0-21-g975b
[torvalds@g5 git]$ git describe --all --abbrev=4 HEAD^ heads/lt/describe-7-g975b
With --abbrev set to 0, the command can be used to find the closest tagname without any suffix:
[torvalds@g5 git]$ git describe --abbrev=0 v1.0.5^2 tags/v1.0.0
Note that the suffix you get if you type these commands today may be longer than what Linus saw above when he ran these commands, as your Git repository may have new commits whose object names begin with 975b that did not exist back then, and "-g975b" suffix alone may not be sufficient to disambiguate these commits.
For each commit-ish supplied, git describe will first look for a tag which tags exactly that commit. Annotated tags will always be preferred over lightweight tags, and tags with newer dates will always be preferred over tags with older dates. If an exact match is found, its name will be output and searching will stop.
If an exact match was not found, git describe will walk back through the commit history to locate an ancestor commit which has been tagged. The ancestor's tag will be output along with an abbreviation of the input commit-ish's SHA-1. If --first-parent was specified then the walk will only consider the first parent of each commit.
If multiple tags were found during the walk then the tag which has the fewest commits different from the input commit-ish will be selected and output. Here fewest commits different is defined as the number of commits which would be shown by git log tag..input will be the smallest number of commits possible.