Syntax tables do for input what compositions do for output: They allow you to teach custom notations to Calc's formula parser. Calc keeps a separate syntax table for each language mode.
(Note that the Calc "syntax tables" discussed here are completely unrelated to the syntax tables described in the Emacs manual.)
The Z S (calc-edit-user-syntax) command edits the
syntax table for the current language mode. If you want your
syntax to work in any language, define it in the normal language
mode. Type M-# M-# to finish editing the syntax table, or
M-# x to cancel the edit. The m m command saves all
the syntax tables along with the other mode settings;
see section General Mode Commands.