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 infrequently, 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 (email@example.com), or contact TUG.
If you’ve found a bug in LaTeX 2e, report it using mechanisms supplied by the LaTeX team. (Please be careful to ensure you’ve got a LaTeX bug, or a bug in one of the “required” packages distributed by the LaTeX team.)
If you’ve found a bug in a contributed LaTeX package, the best starting place is usually to ask in the “usual places for help on-line, or just possibly one of the specialised mailing lists. The author of the package may well answer on-line, but if no-one offers any help, you may stand a chance if you mail the author (presuming that you can find an address…).
If you’ve found a bug in LaTeX 2.09, or some other such unsupported software, your only real hope is help on-line.
Failing all else, you may need to pay for help — TUG maintains a register of TeX consultants. (This of course requires that you have the resources — and a pressing enough need — to hire someone.)