For a start, make entirely sure you have found a bug. Double-check with books about TeX, LaTeX, or whatever you're using; compare what you're seeing against the other answers above; ask every possible person you know who has any TeX-related expertise. The reasons for all this caution are various.
If you've found a bug in TeX itself, you're a rare animal indeed.
Don Knuth is so sure of the quality of his code that he offers real
money prizes to finders of bugs; the cheques he writes are
such rare items that they are seldom cashed. If you
think you have found a genuine fault in TeX itself (or METAFONT, or the
CM fonts, or the TeXbook), don't immediately write to Knuth,
however. He only looks at bugs once or twice a year, and even then
only after they are agreed as bugs by a small vetting team. In the
first instance, contact Barbara Beeton at the AMS
(bnb@math.ams.org), or contact
TUG.
If you've found a bug in LaTeX2e, look in the bugs database to see
if it's already been reported. If not you should submit details of
the bug to the LaTeX3 team. To do this, you should
process the file latexbug.tex with LaTeX (the file is part
of the LaTeX2e distribution. The process will give you instructions
about what to do
with your bug report (it can, for example, be sent to the team by
email). Please be sparing of the team's time; they're
doing work for the good of the whole LaTeX community, and any time
they spend tracking down non-bugs is time not available to write or
debug new code. Details of the whole process, and an interface to the
database, are available via
http://www.latex-project.org/help.html
If you've found a bug in LaTeX 2.09, or some other such unsupported
software, there's not a lot you can do about it. You may find help or
de facto support on a newsgroup such as
comp.tex.tex or on a mailing list such as
texhax@tex.ac.uk, but posting non-bugs to any of these forums
can lay you open to ridicule! Otherwise you need to go out and find
yourself a willing TeX-consultant - TUG maintains a
register of TeX consultants (see
http://www.tug.org/consultants.html).